Write a definite integral that represents the area of the region. (do not evaluate the integral.) y1 = x2 2x 3 y2 = 2x 12
The definite integral that represents the area of the region under the given curves is:
What is the area of the region under a curve?
- By performing a definite integral between the two locations, one can determine the area under a curve between two points.
- Integrate y = f(x) between the limits of a and b to determine the area under the curve y = f(x) between x = a & x = b.
- With the specified limits, integration can be used to calculate this area.
Given:
The points where the two intersect will be given by:
y₁ = y₂
=> x² + 2x + 3 = 2x + 12
=> x² + 3 = 12
=> x² = 9
=> x = ± 3
For x₁ = 3, y₁ = 2(3) + 12 = 18 => (x₁, y₁) = (3, 18)
For x₂ = -3, y₂ = 2(-3) + 12 = 6 => (x₂, y₂) = (-3, 6)
Now, for the area of the region under first curve :
A₁ =
For the area of the region under the second curve :
A₂ =
For the required area of the region bounded by the two curves will be given by:
A = A₂ - A₁ =
A =
Refer to the image for the area so bounded by the curves.
Hence, The definite integral that represents the area of the region under the given curves is: .
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