Question

5. a) Calculate the area under the curve y = x ^ 3 + 4x + 1 from x = - 3 to x = 3

b) Evaluate: integrate x * e ^ x dx from 0 to 4

Answer 1

Solution.

a)

$$y=x^3+4x+1, -3\leq x\leq 3$$

The graph of this function:

$$S=\int_{-3}^3(x^3+4x+1)dx=(\frac{x^4}{4}+4\frac{x^2}{2}+x)|_{-3}^3=\newline =\frac{81}{4}+18+3-\frac{81}{4}-18+3=6$$

Answer. S=6 sq.units.

b)

$$\int_0^4xe^xdx=\{u=x, dv=e^xdx,du=dx, v=e^x \}=\newline =xe^x|_0^4-\int_0^4e^xdx=4e^4-e^x|_0^4=4e^4-e^4+e^0=3e^4+1$$

Answer. $3e^4+1.$