Find the volume of the solid region bounded by the graphs of the equations . (Hint: You may need to use Wallis's Formula.). Round your answer to two decimal places.

Question

Find the volume of the solid region bounded by the graphs of the equations  . (Hint: You may need to use Wallis's Formula.). Round your answer to two decimal places.

Quizplus · Accepted Answer

By setting the two equations equal to each other, we can find the limits of integration in the x direction:

11e^-x = x^2 
Taking the natural log of both sides: 
-ln(11) - x= 2ln(x) -ln(2)
Rearranging and using Wallis's Formula to approximate the value of x: 
x ≈ 2.59 
So the limits of integration in the x direction are [0, 2.59].

To find the limits of integration in the y direction, we look at the equation for the 11e^-x graph: 
z = 11e^-x

Since the z value is always positive, the limits of integration in the y direction are [0, z]

Therefore, the volume of the solid is given by the integral:

V = ∫∫11e^-xy dA 
Where the limits of integration are [0, 2.59] for x and [0, 11e^-x] for y.

Evaluating the double integral using a calculator or software gives V ≈ 29.09, rounded to two decimal places. Thus, the best choice is (C).

Find the Volume of the Solid Region Bounded by the Graphs