Question

Sketch the area between the curve y = (x-4)^2 and the line y = 2x+7. Use a definite integral to find the area between these two curves.

Answer 1

Answer on Question #48981 - Math - Integral Calculus

Let us find the points where two graphs intersect by solving $(x - 4)^2 = 2x + 7$ . This quadratic equation has two solutions $x = 1, x = 9$ (which is also obvious from the picture).

Thus, the area between these two graphs is

$$S = \int_ {1} ^ {9} \left\{(2 x + 7) - (x - 4) ^ {2} \right\} d x = \int_ {1} ^ {9} (- 9 + 1 0 x - x ^ {2}) d x = (- 9 x + 5 x ^ {2} - \frac {x ^ {3}}{3}) \int_ {1} ^ {9} = \frac {2 5 6}{3}.$$

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