Question

Integrate with respect to v :
∫104v−632v2dv
8
58
−85
85

Answer 1

Answer for Question #51612 - Math - Integral Calculus

Question. Calculate the integral:

$$\int (104v - 632v^2) dv$$

Solution. Indefinite integral is

$$\int (104v - 632v^2) dv = \frac{104 \cdot v^2}{2} - \frac{632 \cdot v^3}{3} + C = 52v^2 - \frac{632v^3}{3} + C.$$

The definite integral on the interval $[a, b]$ is

$$\begin{aligned} \int_{a}^{b} (104v - 632v^2) dv &= \left(52b^2 - \frac{632b^3}{3}\right) - \left(52a^2 - \frac{632a^3}{3}\right) = \\ &= 52(b^2 - a^2) - \frac{632(b^3 - a^3)}{3}. \end{aligned}$$

Answer.

$$\int_{a}^{b} (104v - 632v^2) dv = 52v^2 - \frac{632v^3}{3} + C.$$ $$\int_{a}^{b} (104v - 632v^2) dv = 52(b^2 - a^2) - \frac{632(b^3 - a^3)}{3}.$$

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