Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer.

Between y = −x and y = x/2 for x in [0, 6]

Did you follow suggestion and sketch the lines?

I see a triangle with vertices
(0,0), (6,3) and (6,-6)

You certainly would not need Calculus for this
base = 9, height = 6
area = (1/2)(9)(6) = 27

If you have to use Calculus ....
effective height = x/2 - (-x) = (3/2)x
area = ∫(3/2) x dx from 0 to 6
= (3/4)x^2 | from 0 to 6
= (3/4)(36) - 0 = 27

To find the area of the indicated region between the curves y = -x and y = x/2 for x in the interval [0, 6], we can use the concept of definite integration.

Here are the steps to find the area using integration:

1. Graph the given curves: Plot the graphs of y = -x and y = x/2 on a coordinate plane. This will help visualize the region we are working with.

2. Determine the range of x-values: In this case, we are given the interval [0, 6], which means we need to find the area of the region between the two curves for x values ranging from 0 to 6.

3. Identify the region: Determine the section of the graph between the curves y = -x and y = x/2 within the given x-range. This is the region we want to find the area of.

4. Set up the integral: To find the area, integrate the difference of the upper and lower curves with respect to x over the given x-range. In this case, the upper curve is y = x/2 and the lower curve is y = -x. So, the integral will be:
∫[0, 6] [(x/2) - (-x)] dx

5. Solve the integral: Evaluate the integral from step 4 using the fundamental theorem of calculus. This will yield the area of the region between the curves.

6. Use technology to check your answer: You can use graphing software or an online integral calculator to verify your solution.

By following these steps, you will be able to find the area of the region between the curves y = -x and y = x/2 for x in the interval [0, 6].