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Math  /  Calculus

Question5 Points] DETAILS MY NOTES NCSUCALC1 5.1.1.017.
Graph the region bounded by x=∣y−3∣x=|y-3| and (y−3)2=−(x−12)(y-3)^{2}=-(x-12).
Find its area. □\square

Studdy Solution

STEP 1

1. The problem involves graphing and finding the area of a region bounded by two curves.
2. The equations given are x=∣y−3∣ x = |y - 3| and (y−3)2=−(x−12) (y - 3)^2 = -(x - 12) .
3. We need to understand the shapes and intersections of these curves to find the bounded region.

STEP 2

1. Analyze and graph the equation x=∣y−3∣ x = |y - 3| .
2. Analyze and graph the equation (y−3)2=−(x−12) (y - 3)^2 = -(x - 12) .
3. Determine the points of intersection of the curves.
4. Calculate the area of the region bounded by these curves.

STEP 3

Analyze and graph the equation x=∣y−3∣ x = |y - 3| .
- This equation represents two lines: x=y−3 x = y - 3 and x=−(y−3) x = -(y - 3) . - Rewrite them as: - x=y−3 x = y - 3 or y=x+3 y = x + 3 - x=−(y−3) x = -(y - 3) or y=−x+3 y = -x + 3 - These are two lines intersecting at the point where y=3 y = 3 .

STEP 4

Analyze and graph the equation (y−3)2=−(x−12) (y - 3)^2 = -(x - 12) .
- Rewrite the equation as (y−3)2=−(x−12) (y - 3)^2 = -(x - 12) . - This is a sideways parabola opening to the left with vertex at (12,3) (12, 3) .

STEP 5

Determine the points of intersection of the curves.
- Set x=∣y−3∣ x = |y - 3| equal to the rearranged form of the parabola equation. - Solve for y y when x=y−3 x = y - 3 and x=−(y−3) x = -(y - 3) with (y−3)2=−(x−12) (y - 3)^2 = -(x - 12) . - Find the intersection points by substituting x=y−3 x = y - 3 and x=−(y−3) x = -(y - 3) into the parabola equation.

STEP 6

Solve the system of equations to find the intersection points.
- For x=y−3 x = y - 3 , substitute into (y−3)2=−(x−12) (y - 3)^2 = -(x - 12) : - (y−3)2=−(y−3−12) (y - 3)^2 = -(y - 3 - 12) - Solve for y y .
- For x=−(y−3) x = -(y - 3) , substitute into (y−3)2=−(x−12) (y - 3)^2 = -(x - 12) : - (y−3)2=−(−y+3−12) (y - 3)^2 = -(-y + 3 - 12) - Solve for y y .

STEP 7

Calculate the area of the region bounded by these curves.
- Use integration or geometric methods to find the area between the curves. - Integrate the difference of the functions over the interval determined by the intersection points.
The area of the region is calculated value\boxed{\text{calculated value}}.

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