Question22. Consider and . (1) Find the points of intersection, (2) Find the area of the region between two curves.
Studdy Solution
STEP 1
Assumptions
1. The function represents a parabola.
2. The function represents a straight line.
3. We need to find the points of intersection of these two functions.
4. We need to calculate the area of the region between the two curves over the interval defined by their points of intersection.
STEP 2
To find the points of intersection, set .
STEP 3
Rearrange the equation to form a quadratic equation.
STEP 4
Solve the quadratic equation using the quadratic formula:
where , , and .
STEP 5
Calculate the discriminant .
STEP 6
Substitute the values into the quadratic formula.
STEP 7
Calculate the two possible solutions for .
STEP 8
The points of intersection are when and . Find the corresponding -values using .
For , .
For , .
Thus, the points of intersection are and .
STEP 9
To find the area between the curves, integrate the difference from to .
STEP 10
Integrate the function .
STEP 11
Evaluate the definite integral from to .
STEP 12
Calculate the value at .
STEP 13
Calculate the value at .
STEP 14
Subtract the result at from the result at .
STEP 15
Since the area cannot be negative, take the absolute value.
The area of the region between the two curves is square units.
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