Question

Help, please !!! Answer all questions please. Thank u so much!!!

8 Given the graph of f(x) below find f(x)dx. 71 6 5 4 3 2 1 6 8 9 - 1 -2 Find the exact value and no decimal entries. f(x)dx = 1

Answer 1

The integration is given by the area under the curve and the sign is positive if the curve is above the x-axis and negative if curve is below the x-axis.

$\small \int_{-1}^{8}f(x)dx=\int_{-1}^{3}f(x)dx+\int_{3}^{7}f(x)dx+\int_{7}^{8}f(x)dx$

=(Area of triangle with h=4, b=4) - (Area of semi-circle with dia.=4)+(Area of triangle h=2, b=1)

  $\small =(\frac{1}{2}\times 4\times 4)-\frac{\pi(\frac{4}{2})^2}{2}+\frac{1}{2}\times 2\times 1$$\small (\because area\: of\: triangle=\frac{1}{2}\times b\times h)$

  $\small =8-2\pi+1$

$\small \Rightarrow\mathbf{ \int_{-1}^{8}f(x)dx=9-2\pi}$