Area between Curves
1. Find the area of the region under the graph of \(f\) on \( [a, b]\).
solution
(a) \( f(x)=x^2-2x+2;\quad [-1, 2] \)
(b) \(\displaystyle{f(x)= \frac{1}{x^2};\quad [1, 2] }\)
2. Use a definite integral to find the area under the curve between the given x-values. solution
\[f(x)=8x^3, \quad x=2 \: \text{to} \: x=3\]
3. Use a definite integral to find the area under the curve between the given x-values. solution
\[f(x)=\frac{1}{\sqrt{x}}, \quad x=9 \: \text{to} \: x=16\]
4. Find the area between the curves \( f(x)=x \) and \(
g(x)=x^2\). solution
![graph](g1.png)
5. Find the area between the curves \( y=2x-1 \) and \(
y=x^2-1\). solution
![graph2](g2.PNG)
3. Find the area between the curves as shown.
solution
![graph3](g3.PNG)