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

5. Find the area between the curves \( y=2x-1 \) and \( y=x^2-1\).  solution

3. Find the area between the curves as shown.  solution