Answer to Question #235616 in Calculus for john k

Question

4)

Find the area of the region bounded by the curves

Y = X² - 4X + 4 and Y = 10 - X²

From X = 10 and X = 4

Answer 1

We have to process the integral of area $A= \int_{4}^{10} \Big[ f(x)-g(x) \Big] {dx};$ we also define

$\\ f(x)=x^2-4x+4 \text{ (we define this function as the superior function)} \\ g(x)=10-x^2 \text{ (we define this function as the inferior function)}$

Then we substitute and proceed to solve the integral and find the value for the area:

$A= \int^{10}_{4} \Big[ f(x)-g(x) \Big] {dx} \\ A= \int_{4}^{10} \Big[ (x^2-4x+4)-(10-x^2) \Big] {dx} \\ A= \int_{4}^{10} \Big( 2x^2-4x -6 \Big) {dx} \\A= 2 \int_{4}^{10} \Big( x^2-2x-3 \Big) {dx}$

We proceed to solve the integral and substitute the limits to find the area:

$\\A=2\bigg[ \cfrac{x^3}{3}-x^2-3x \bigg]_{4}^{10} \\A=2\bigg\{ \Big[\cfrac{(10^3)}{3}- 3(10)-(10^2) \Big] - \Big[ \cfrac{(4^3)}{3}-3(4)-(4^2) \Big] \bigg\} \\A=2\bigg\{\cfrac{1000}{3}-30-100- \Big(\cfrac{64}{3} -12-16 \Big) \bigg\} \\ A=2 \big( 210 \big) \\ \implies A =420 \text{ area units (u}^2)$

In conclusion, the area defined of the region boundedby the curves Y = X² - 4X + 4 and Y = 10 - X² from X = 4 and X = 10 is equal toA= 420 u² (the solution will have area units).Reference:

• Thomas, G. B., & Finney, R. L. (1961). Calculus. Addison-Wesley Publishing Company.