Evaluate the following definite integrals using the Fundamental Theorem of Calculus $$ \int_{-10}^1 s | 25 - s^2 | \; \mathrm d s. $$
my work:
$$ s=\pm 5 $$
$$ \int^{-5}_{-10} f(s) + \int^5_{-5} f(s) + \int^1_5 f(s) $$
Stuck here. Can't move to next step. Help please
$\endgroup$3 Answers
$\begingroup$HINT:
Write the integral as
$$\int_{-10}^1s|25-s^2|\,ds=\int_{-10}^{-5}s(s^2-25)\,ds+\int_{-5}^1s(25-s^2)\,ds$$
Can you finish from here?
$\endgroup$ 0 $\begingroup$Hint: See what is the sign of $25-s^2$ in each of the intervals for $s$ and write down the module $|25-s^2|$
$\endgroup$ $\begingroup$Hint: $ |25-s²| = \begin{cases} 25-s^2, \text{ if } 25-s^2 \geq 0 \therefore -5 \leq s \leq 5 \\ s^2 - 25, \text{ if } 25-s^2 < 0 \therefore s < -5 \text{ or } s > 5 \end{cases} $, so the definite integral can be rewritten as:
$ \int_{-10}^1 s \cdot |25-s^2| \,\, ds = \int _{-10}^{-5} s^3 - 25s \,\, ds + \int _{-5}^{1} -s^3 + 25s \,\, ds $
$\endgroup$
