QuestionFind the area of the shaded region.
Studdy Solution
STEP 1
1. We are given two functions: and .
2. The shaded region is the area between these two curves.
3. We need to find the points of intersection to determine the limits of integration.
4. The area of the shaded region is found by integrating the difference between the two functions over the interval determined by the points of intersection.
STEP 2
1. Find the points of intersection of the functions and .
2. Determine the limits of integration from the points of intersection.
3. Set up the integral of the difference over the interval.
4. Evaluate the integral to find the area of the shaded region.
STEP 3
Find the points of intersection by setting :
STEP 4
Rearrange the equation to set it to zero:
STEP 5
Solve the polynomial equation for to find the points of intersection. This may require numerical methods or graphing to approximate solutions if they are not easily factorable.
STEP 6
Determine the limits of integration from the -values of the points of intersection found in Step 3.
STEP 7
Set up the integral for the area of the shaded region:
where and are the limits of integration determined in Step 4.
STEP 8
Substitute the functions into the integral:
STEP 9
Evaluate the integral to find the area:
Calculate the definite integral by substituting the limits and .
The solution will be the numerical value obtained after evaluating the definite integral.
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