How do you determine the area enclosed by an ellipse #x^2/5 + y^2/ 3# using the trapezoidal rule?

Answer 1

Assuming that the ellipse is #x^2/5+y^2/3 = 1#, see the explanation section below.

The center of this ellipse is on the #x# axis, so the area of the ellipse is twice the area of the upper half of the ellipse.
Solve #x^2/5+y^2/3 = 1# for #y# to get the upper half of the ellipse.

The upper half of the ellipse is described by the function

#y = f(x) = sqrt((15-3x^2)/5)#.
Note that the values of #x# vary from #-sqrt5# to #sqrt5#.
To find the area using the trapezoidal rule we need an positive integer #n#. Since none is given we'll just go through the general method of solution.
#{a,b] = [-sqrt5, sqrt5]#, and .
So we get #Delta x = (b-a)/n = (sqrt5-(-sqrt5))/n = (2sqrt5)/n#
The endpoints of the subintervals are found by beginning at #a=-sqrt5# and successively adding #Delta x = (2sqrt5)/n# to find the points until we get to #x_n = b = pi#.
#x_0 = -sqrt5#, #x_1 = -sqrt5 + (2sqrt5)/n#, #x_2 = -sqrt5 + 2(2sqrt5)/n#, . . . , #x_i = -sqrt5 + i(2sqrt5)/n#, . . . , and #x_n = -sqrt5 + n(2sqrt5)/n= sqrt5 = b#
Now apply the formula (do the arithmetic) for #f(x) = sqrt((15-3x^2)/5)#..
#T_n=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * +2f(x_(n-1))+f(x_n)] #
The total area of the ellipse is #~~2T_n#
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Answer 2

To determine the area enclosed by the ellipse ( \frac{x^2}{5} + \frac{y^2}{3} = 1 ) using the trapezoidal rule, we first need to approximate the curve by dividing it into small segments and approximating each segment as a trapezoid.

  1. Divide the ellipse into ( n ) equal segments along the x-axis.
  2. Calculate the corresponding y-values for each segment using the equation of the ellipse: ( y = \sqrt{3(1 - \frac{x^2}{5})} ).
  3. Approximate each segment as a trapezoid by connecting consecutive points with straight lines.
  4. Calculate the area of each trapezoid using the formula for the area of a trapezoid: ( A = \frac{1}{2}(b_1 + b_2)h ), where ( b_1 ) and ( b_2 ) are the lengths of the parallel sides and ( h ) is the height (the width of each segment along the x-axis).
  5. Sum up the areas of all trapezoids to obtain an approximation of the total area enclosed by the ellipse.

The more segments you use (i.e., the larger the value of ( n )), the more accurate your approximation will be.

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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.

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