Question

Area under the curve problems:

a. y = x, y = x^3
b. x = y^2 + 1, x = y + 3
c. y = csc(x), y = 2, π/6 ≤ x ≤ 5π/6
d. y = sin(x), y = cos(x), π/4 ≤ x

Answer 1

To find the area under the curve between two functions, you need to determine the points of intersection between the two functions and integrate the difference between them over the given interval.

Step-by-step explanation:

To find the area under the curve between two functions, you need to determine the points of intersection between the two functions and integrate the difference between them over the given interval. Let's consider each problem:

a. For the functions y = x and y = x^3, the area under the curve can be found by integrating the difference between the two functions over the interval where they intersect.

b. For the functions x = y^2 + 1 and x = y + 3, find the points of intersection and integrate the difference between the two functions over the given interval.

c. For the functions y = csc(x) and y = 2, integrate the difference between the two functions over the interval π/6 ≤ x ≤ 5π/6.

d. For the functions y = sin(x) and y = cos(x), integrate the difference between the two functions over the interval π/4 ≤ x.