How do you find the area bounded by #y=x#, #y=1/x^2# the x axis, and x=3?

Answer 1

#S = 7/6#

Start by solving the inequality:

#x <= 1/x^2 => x^3 <= 1 => x < 1#

so we can see that in the interval #[0,1]# the area is bounded by the curve #y = x# while in the interval #[1,3]# the area is bounded by the curve #y=1/x^2#

The required area can be calculated as the definite integral of the bounding curve and therefore is:

#S = int_0^1 xdx + int_1^3 dx/x^2#

#S = [x^2/2]_0^1 - [1/x]_1^3#

#S = 1/2 - 0 -1/3 + 1 = 7/6#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the area bounded by the curves ( y = x ), ( y = \frac{1}{x^2} ), the x-axis, and ( x = 3 ), you need to integrate the absolute difference between the two curves from their intersection points to ( x = 3 ).

First, find the intersection point by setting the two equations equal to each other: ( x = \frac{1}{x^2} ). Solve for ( x ) to get ( x^3 = 1 ), which gives ( x = 1 ).

Then, integrate the absolute difference between the two functions from ( x = 1 ) to ( x = 3 ):

[ \text{Area} = \int_{1}^{3} \left| (x) - \left( \frac{1}{x^2} \right) \right| , dx ]

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To find the area bounded by ( y = x ), ( y = \frac{1}{x^2} ), the x-axis, and ( x = 3 ):

  1. Determine the points of intersection of the curves ( y = x ) and ( y = \frac{1}{x^2} ) by setting them equal to each other and solving for ( x ).

  2. Identify the region of interest based on these points of intersection and the given boundaries.

  3. Integrate the difference of the upper curve and the lower curve within the specified bounds to find the area.

[ \text{Area} = \int_{a}^{b} (f(x) - g(x)) , dx ]

Where ( f(x) ) is the upper curve and ( g(x) ) is the lower curve within the given bounds ([a, b]).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7