How do you find the area in the first quadrant bounded by #y=x^-2# and #y=17/4 - x^2#?

Question

Cameron Alexander · Accepted Answer

The area is #9/4# square units.

Start by finding the points of intersection.

#x^(-2) = 17/4 - x^2#
#1/x^2 = 17/4 - x^2#
#1/x^2 + x^2 - 17/4= 0#
#4x^4 - 17x^2 + 4 = 0#

Now let #u = x^2#.

#4u^2 - 17u + 4 = 0#
#4u^2 - 16u - u + 4 = 0#
#4u(u - 4) - (u - 4) = 0#
#(4u - 1)(u - 4) = 0#
#u = 1/4 or 4#
#x^2 = 1/4 or x^2 = 4#
#x = +- 1/2 or x = +-2#

Since we're talking about the area exclusively in the first quadrant, our standard integral #int_a^b# becomes #int_(1/2)^2#. Now draw a rudimentary graph of the two functions.

The graph in #color(blue)("blue")# is the parabola #y = 17/4 - x^2# and the graph in #color(red)("red")# is the rational function #y = x^(-2)#. As you can see, the parabola is above the rational function, therefore our expression that gives us the area will be

#A = int_(1/2)^2 17/4 - x^2 - (x^-2) dx#
#A = [17/4(x) - 1/3x^3 + x^-1]_(1/2)^2#

This can be evaluated using the second fundamental theorem of calculus which states that #int_a^b F(x) dx = f(b) - f(a)#, where #F(x)# is continuous on #[a, b]# and #f'(x) = F(x)#.

#A = 17/4(2) - 1/3(2)^3 + 2^-1 - (17/4(1/2) - 1/3(1/2)^3 + (1/2)^-1)#
#A = 17/2 - 8/3 + 1/2 - 17/8 + 1/24 - 2#
#A = 7 - 8/3 - 50/24#
#A = 54/24 = 9/4 "square units"#

Hopefully this helps!

Isaiah Allen · Answer

undefined ToTo findTo find the area in the first quadrant bounded by $y = xTo find the area in the first quadrant bounded by \(y = x^{-To find the area in the first quadrant bounded by \(y = x^{-2}\To find the area in the first quadrant bounded by \(y = x^{-2}$ andTo find the area in the first quadrant bounded by $y = x^{-2}$ and $yTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \fracTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{To find the area in the first quadrant bounded by \( yTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17To find the area in the first quadrant bounded by \( y = x^{-To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{To find the area in the first quadrant bounded by \( y = x^{-2}To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4To find the area in the first quadrant bounded by \( y = x^{-2} \To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4}To find the area in the first quadrant bounded by \( y = x^{-2} $To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} -To find the area in the first quadrant bounded by \( y = x^{-2} $ andTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - xTo find the area in the first quadrant bounded by \( y = x^{-2} $ and $To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^To find the area in the first quadrant bounded by \( y = x^{-2} $ and $ yTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2To find the area in the first quadrant bounded by \( y = x^{-2} $ and $ y =To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$,To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow theseTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \fracTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these stepsTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4}To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} -To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1.To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - xTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersectionTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection byTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by settingTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $,To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting theTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, followTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equationsTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow theseTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal toTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these stepsTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other andTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $xTo find the area in the first quadrant bounded by \( y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x\To find the area in the first quadrant bounded by \( y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the twoTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equalTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal toTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2.To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to eachTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. IntTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each otherTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. IntegrateTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other andTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the differenceTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve forTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of theTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upperTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ xTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper andTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x \To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lowerTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functionsTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ toTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions withTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to findTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respectTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find theTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $To find the area in the first quadrant bounded by \( y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points ofTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $xTo find the area in the first quadrant bounded by \( y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersectionTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ overTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over theTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the intervalTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2.To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval whereTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. DetermineTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where theyTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine theTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersectTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limitsTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits ofTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3To find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integrationTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take theTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration forTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute valueTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding theTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute value ofTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding the $To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute value of theTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding the $ xTo find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute value of the resultTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding the $ x \To find the area in the first quadrant bounded by \(y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute value of the result to ensureTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding the $ x $-To find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute value of the result to ensure aTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding the $ x $-valuesTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute value of the result to ensure a positiveTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding the $ x $-values whereTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps:

1. Find the points of intersection by setting the equations equal to each other and solving for $x$.
2. Integrate the difference of the upper and lower functions with respect to $x$ over the interval where they intersect.
3. Take the absolute value of the result to ensure a positive areaTo find the area in the first quadrant bounded by $ y = x^{-2} $ and $ y = \frac{17}{4} - x^2 $, follow these steps:

1. Set the two equations equal to each other and solve for $ x $ to find the points of intersection.
2. Determine the limits of integration for $ x $ by finding the $ x $-values where theTo find the area in the first quadrant bounded by $y = x^{-2}$ and $y = \frac{17}{4} - x^2$, follow these steps: