Question

Set up an integral in terms of x that can be used to find the area of the region bounded by the given curves.

a) ∫a, b y(x) dx
b) ∫c, d x(y) dy
c) ∫e, f √(1 + (dy/dx)^2) dx
d) ∫g, h √(1 + (dx/dy)^2) dy

Answer 1

To find the area of a region bounded by curves, you can set up integrals in terms of x or y, or using the square root of the derivative.

Step-by-step explanation:

To set up an integral to find the area of the region bounded by the given curves:

a) For the integral in terms of x, you need to integrate the function y(x) with respect to x. So, the integral is: ∫ab y(x) dx

b) For the integral in terms of y, you need to integrate the function x(y) with respect to y. So, the integral is: ∫cd x(y) dy

c) For the integral involving the square root of the derivative, you need to find the derivative dy/dx and use it to set up the integral. The integral is: ∫ef √(1 + (dy/dx)^2) dx

d) For the integral involving the square root of the derivative, you need to find the derivative dx/dy and use it to set up the integral. The integral is: ∫gh √(1 + (dx/dy)^2) dy