ASAP USE A CALCULATOR AND CLEARLY WRITE ANSWER

Question

Let the region R be the area enclosed by the function f(x)=e^x and g(x)=5x+1. If the region R is the base of a solid such that each cross-section perpendicular to the x-axis is a square, find the volume of the solid. You may use a calculator and round to the nearest thousandth.

Answer 1

_{Limits of integration: e}^x = 5x + 1; x = 0 or x = 2.66013991

Volume V = ∫₀^2.66013991(5x + 1 - e^x)²dx = 22.66143043

To get x-coordinate and evaluate integral we used TI-84

Answer 2

f(x) is convex and g(x) is linear

f(0) = g(0) = 1

f'(0) = 1

g'(0) = 5

Thus, the curves meet at 0 and at some point greater than 0.

As e^2 < 5(2) + 1 and e^3 > 5(3) + 1, we know the other solution is between 2 and 3.

The bisection method gave us a solution of 2.66039905846446

Using 0 and 2.66039905846446, we used Simpson's rule with 1,000,000 intervals and got a value of the area under (g(x) - f(x))^2 on this interval to be 22.661

We also could have easily integrated the function on this interval.