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.
A) 9.26
B) 32.21
C) 29.09
D) 58.18
E) 18.52

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.
A) 9.26 
B) 32.21 
C) 29.09 
D) 58.18 
E) 18.52

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