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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 \(y = 4\).