Question

let R be the region bounded by the graphs of y = sin(pie times x) and y = x^3 - 4.

a) find the area of R

b) the horizontal line y = -2 splits the region R into parts. write but do not evaluate an integral expression for the area of the part of R that is below this horizontal line.

c) The region R is the base of a solid. For this solid, each cross section perpendicular to x-axis is a square. Find the volume of this solid.

d) the region R models the surface of a small pond. At all points in R at a distance x from the y-axis, the depth of the water is given by h(x)=3-x. find the volume of the water in the pound.

Answer 1

A. Area is the Integral of (sin(pi*X)-(X^3-4X)) from 0 to 2 which should equal 4.
B. The horizontal line y=-2 intersects the graph of X^3-4X at the points .539 and 1.675. The area is going to be the Integral of (sin(pi*X)-(X^3-4X)) from .539 to 1.675
C. Volume= the integral of (sin(pi*X)-(X^3-4X))^2 from 0 to 2 because the height is proportional to the width of the base. this should equal 9.978