# Section 3.1 Calculus 2

Area Between Curves

## 3.1 Area Between Curves

### 3.1.1 Areas between Functions of $$x$$

• Recall that $$\int_a^b f(x)\,dx$$ is the net area between $$y=f(x)$$ and $$y=0$$.
• Let $$f(x)\leq g(x)$$ for $$a\leq x\leq b$$. We define the area between the curves $$y=f(x)$$ and $$y=g(x)$$ from $$a$$ to $$b$$ to be the integral $$\int_a^b [g(x)-f(x)]\,dx$$.
• We call $$y=f(x)$$ the bottom curve and $$y=g(x)$$ the top curve.
• Example Find the area between the curves $$y=2+x$$ and $$y=1-\frac{1}{2}x$$ from $$2$$ to $$4$$.
• Example Find the area bounded by the curves $$y=x^2-4$$ and $$y=8-2x^2$$.
• Example Prove that the area of a circle of radius $$r$$ is $$\pi r^2$$. (Hint: use the curves $$y=\pm\sqrt{r^2-x^2}$$.)

### 3.1.2 Areas between Functions of $$y$$

• Areas between functions $$f(y)\leq g(y)$$ may be found similarly, but in this case $$x=f(y)$$ is the left curve and $$x=g(y)$$ is the right curve.
• Example Find the area bounded by the curves $$y=\sqrt{x}$$, $$y=0$$, and $$y=x-2$$.

### Review Exercises

1. Find the area between the curves $$y=4$$ and $$y=4x^3$$ from $$-1$$ to $$1$$.
2. Find the area bounded by the curves $$y=x^2-2x$$ and $$y=x$$.
3. Find the area bounded by the curves $$y=\pm\sqrt{4-x}$$ and $$x=3$$.
4. Find the area bounded by the curves $$y=0$$, $$x=0$$, $$y=1$$, and $$y=\ln x$$.
5. Use a definite integral to prove that the area of the triangle with vertices $$(0,0)$$, $$(b,0)$$, $$(0,h)$$ is $$\frac{1}{2}bh$$.
6. Find the area of the ellipse $$9x^2+16y^2=25$$.

Solutions

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 5.6