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### Section 6-2 : Area Between Curves

1. Determine the area below $$f\left( x \right) = 8x - 2{x^2}$$ and above the x-axis.
2. Determine the area above $$f\left( x \right) = 3{x^2} + 6x - 9$$ and below the x-axis.
3. Determine the area to the right of $$g\left( y \right) = {y^2} + 4y - 5$$ and to the left of the y-axis.
4. Determine the area to the left of $$g\left( y \right) = - 4{y^2} + 24y - 20$$ and to the right of the y-axis.
5. Determine the area below $$f\left( x \right) = 10 - 2{x^2}$$ and above the line $$y = 3$$.
6. Determine the area above $$f\left( x \right) = {x^2} + 2x + 3$$ and below the line $$y = 11$$.
7. Determine the area to the right of $$g\left( y \right) = {y^2} + 2y - 4$$ and to the left of the line $$x = - 1$$.
8. Determine the area to the left of $$g\left( y \right) = 2 + 4y - {y^2}$$ and to the right of the line $$x = - 1$$.

For problems 9 – 26 determine the area of the region bounded by the given set of curves.

1. $$y = {x^3} + 2$$, $$y = 1$$ and $$x = 2$$.
2. $$y = {x^2} - 6x + 10$$ and $$y = 5$$.
3. $$y = {x^2} - 6x + 10$$, $$x = 1$$, $$x = 5$$ and the x-axis.
4. $$x = {y^2} + 2y + 4$$ and $$x = 4$$.
5. $$y = 5 - \sqrt x$$, $$x = 1$$, $$x = 4$$ and the x-axis.
6. $$x = {{\bf{e}}^y}$$, $$x = 1$$, $$y = 1$$ and $$y = 2$$.
7. $$x = 4y - {y^2}$$ and the y-axis.
8. $$y = {x^2} + 2x + 4$$, $$y = 3x + 6$$, $$x = - 3$$ and $$x = 3$$.
9. $$x = 6y - {y^2}$$, $$x = 2y$$, $$y = - 2$$ and $$y = 5$$.
10. $$y = {x^2} + 8$$, $$y = 3{x^2}$$, $$x = - 3$$ and $$x = 4$$.
11. $$x = {y^2}$$, $$x = {y^3}$$ and $$y = 2$$.
12. $$\displaystyle y = \frac{7}{x}$$, $$\displaystyle y = \frac{1}{x} - 3$$, $$x = - 1$$and $$x = - 4$$.
13. $$y = 2{x^2} + 1$$, $$y = 7 - x$$, $$x = 4$$ and the y-axis.
14. $$\displaystyle y = \sin \left( {\frac{1}{2}x} \right)$$, $$y = 3 + \cos \left( {2x} \right)$$, $$x = 0$$ and $$x = \frac{\pi }{4}$$.
15. $$x = \sqrt {2y + 6}$$, $$x = y - 1$$, $$y = 1$$ and $$y = 6$$.
16. $$y = 2 - {{\bf{e}}^{2 - x}}$$, $$y = {x^2} - 4x + 7$$, $$x = 3$$ and the y-axis. Note : These functions do not intersect.
17. $$y = {{\bf{e}}^{2x - 1}}$$, $$y = {{\bf{e}}^{5 - x}}$$, $$x = 0$$ and $$x = 3$$.
18. $$x = \cos \left( {\pi y} \right)$$, $$x = 3$$, $$y = 0$$ and $$y = 4$$.