Question

2. Geometric interpretation of integrals. Consider the integral where R is the region bounded by the a-axis, p-axis and r +y- 2 (a) Let =-z-v + 2, what object does this equation (NOT the integral) represent? (b) Interpret the integral as the volume of a shape. Sketch the shape. (e) Compute the integral by computing the volume of the shape. Page 3

Answer 1

$z=-x-y+2$ represents a surface.

5 0 5 5 5 2 2101

The Double Integral of $z=-x-y+2$ in the region bounded by x-axis y-axis and $x+y=2$ is the volume of the solid shown in the figure.

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The volume of the shape is given by

$\int _0^2\int _0^{2-x}-x-y+2dydx=\frac{4}{3}$

$\int _0^2\int _0^{2-x}-x-y+2dydx=\int _0^2(\frac{x^2-4x-4}{2}+4)dx$

$\int _0^2\left(\frac{x^2-4x-4}{2}+4\right)dx=\frac{4}{3}$