Question

Find the area of the region that lies under the parabola y=5x - x^2, where 1≤x≤4. I would definitely like to see some work :)

Answer 1

16.5 square units

Explanation:

You are expected to integrate the function between x=1 and x=4:

`\displaystyle\text{area}=\int_1^4{(5x-x^2)}\,dx=\left.\left((5)/(2)x^2-(1)/(3)x^3\right)\right|_(x=1)^(x=4)\\\\=(5(4^2-1^2))/(2)-(4^3-1^3)/(3)=37.5-21=\boxed{16.5}`

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Additional comment

If you're aware that the area inside a (symmetrical) parabola is 2/3 of the area of the enclosing rectangle, you can compute the desired area as follows.

The parabolic curve is 4-1 = 3 units wide between x=1 and x=4. It extends upward 2.25 units from y=4 to y=6.25, so the enclosing rectangle is 3×2.25 = 6.75 square units. 2/3 of that area is (2/3)(6.75) = 4.5 square units.

This region sits on top of a rectangle 3 units wide and 4 units high, so the total area under the parabolic curve is ...

area = 4.5 +3×4 = 16.5 . . . square units