Question

Evaluate the integral.

Answer 1

`\bf \displaystyle \int\limits_(-4)^4\ \left[ f(x)= \begin{cases} 2&-4\le x\le 0\\ 0&\ \textless \ x\le 4 \end{cases} \right] \cdot dx\\\\ -------------------------------\\\\`


`\bf \displaystyle \int\limits_(-4)^0\ 2\cdot dx+\int\limits_(0)^4\ (5-x^2)\cdot dx \\\\\\ \displaystyle \int\limits_(-4)^0\ 2\cdot dx+\int\limits_(0)^4\ 5\cdot dx-\int\limits_(0)^4\ x^2\cdot dx\implies \left. 2x \cfrac{}{} \right]_(-4)^0 +\quad \left. 5x \cfrac{}{}\right]_(0)^4-\quad \left. \cfrac{x^3}{3}\right]_(0)^4 \\\\\\ 8+20-\cfrac{64}{3}\implies \cfrac{20}{3}`