CALCULUS NEED HELP ASAP PLEASE THANK YOU!

Question

g(x) = x². Find the area bounded by f(x), g(x) and the x-axis in the interval [0.5, 1]. Your work must show the integral, but you may use your calculator to evaluate it. Give 3 decimal places in your final answer.

Answer 1

Since you don't discuss f(x), I'm going to assume that f(x) is in a different problem.

Here is a sketch:

The area is the area under g(x) between 0.5 and 1 which is the integral as shown.

To evaluate the integral, take the antiderivative which would be x³/3 (because, when you take the derivative of x³/3, you get x²) and we evaluate the antiderivative at the two endpoints and subtract like this:

A = (1)³/3 - (0.5)³/3 = 1/3 - 1/24 = 8/24 - 1/24 = 7/24 square units or (using a decimal approximation) A = 0.292 square units