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### Section 6-2 : Area Between Curves

1. Determine the area below $$f\left( x \right) = 3 + 2x - {x^2}$$ and above the x-axis. Solution
2. Determine the area to the left of $$g\left( y \right) = 3 - {y^2}$$ and to the right of $$x = - 1$$. Solution

For problems 3 – 11 determine the area of the region bounded by the given set of curves.

1. $$y = {x^2} + 2$$, $$y = \sin \left( x \right)$$, $$x = - 1$$ and $$x = 2$$ Solution
2. $$\displaystyle y = \frac{8}{x}$$, $$y = 2x$$ and $$x = 4$$ Solution
3. $$x = 3 + {y^2}$$, $$x = 2 - {y^2}$$, $$y = 1$$ and $$y = - 2$$ Solution
4. $$x = {y^2} - y - 6$$ and $$x = 2y + 4$$ Solution
5. $$y = x\sqrt {{x^2} + 1}$$, $$y = {{\bf{e}}^{ - \,\,\frac{1}{2}x}}$$, $$x = - 3$$ and the y-axis. Solution
6. $$y = 4x + 3$$, $$y = 6 - x - 2{x^2}$$, $$x = - 4$$ and $$x = 2$$ Solution
7. $$\displaystyle y = \frac{1}{{x + 2}}$$, $$y = {\left( {x + 2} \right)^2}$$, $$\displaystyle x = - \frac{3}{2}$$, $$x = 1$$ Solution
8. $$x = {y^2} + 1$$, $$x = 5$$, $$y = - 3$$ and $$y = 3$$ Solution
9. $$x = {{\bf{e}}^{1 + 2y}}$$, $$x = {{\bf{e}}^{1 - y}}$$, $$y = - 2$$ and $$y = 1$$ Solution