# Section 3.2 Calculus 2

Volumes by Cross-Sectioning

## 3.2 Volumes by Cross-Sectioning

### 3.2.1 Defining Volume with Integrals

• The volume of a solid defined between $$x=a$$ to $$x=b$$ with a cross-sectional area of $$A(x)$$ at each $$x$$-value is defined to be $$V=\int_a^b A(x)\,dx$$.
• Steps for solving such problems:
1. Sketch the solid along the $$x$$-axis with a typical cross-section at some $$x$$ value.
2. Find the formula for $$A(x)$$, and the minimal/maximal $$x$$ values $$a,b$$.
3. Evaluate $$V=\int_a^b A(x)\,dx$$.
• Example Show that the volume of a pyramid with a square base of sidelength $$2$$ and height $$3$$ is $$4$$ cubic units.

### 3.2.2 Circular Cross-Sections

• In the case that all cross-sections are circular, we may replace $$A(x)$$ with $$\pi[R(x)]^2$$, where $$R(x)$$ is the radius of the circular cross-section at that $$x$$ value.
• Example Prove that a cone of radius $$r$$ and height $$h$$ has volume $$V=\frac{1}{3}\pi r^2 h$$.

### Review Exercises

1. Find the volume of a solid located between $$x=-1$$ and $$x=2$$ with cross-sectional area $$A(x)=x^2+1$$ for all $$-1\leq x\leq 2$$.
2. Find the volume of a solid located between $$x=0$$ and $$x=1$$ whose cross-sections are parallelograms with base length $$b(x)=x+1$$ and height $$h(x)=x^2+1$$ for all $$0\leq x\leq 1$$.
3. Find the volume of a wedge cut from a circular cylinder with radius $$2$$, sliced out at a $$45^\circ$$ angle from the diameter of its base. (Hint: Sketch the diameter of the cylinder along the $$x$$-axis from $$-2$$ to $$2$$, and use the equation $$x^2+y^2=2^2$$. The cross-sections will be isosceles triangles.)
4. Prove that the volume of a sphere with radius $$r$$ is $$V=\frac{4}{3}\pi r^3$$. (Hint: Draw a diameter of the sphere on the $$x$$-axis from $$-r$$ to $$r$$, and use the equation $$x^2+y^2=r^2$$.)
5. Find the volume of the solid whose base is the region $$0\leq y\leq 4-x^2$$ and whose cross-sections are equilateral triangles perpendicular to the $$x$$-axis.

Solutions

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 6.1