# Calculate the area of the shape bounded by the lines y = x ^ 2-4x + 5 and y = 5.

July 28, 2021 | education

| Let’s find the points of intersection of the lines analytically:

x ^ 2 – 4x + 5 = 5,

x (x – 4) = 0,

x = 0, x = 4

Lower limit of integration: x = 0

Upper limit of integration: x = 4

The desired figure is bounded from above by the straight line y = 5, and from below by the parabola y = x ^ 2-4x + 5, then:

(5 – x ^ 2 + 4x + 5) dx = (- x ^ 2 + 4x) dx = -x ^ 3/3 + 2x ^ 2

Since the boundaries are equal to x = 0, x = 4, respectively, we substitute only the upper limit, because the lower one gives zero, into the resulting expression:

-64/3 + 32 = 32/3

Answer: the area of the figure is 32/3

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