Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Definite integrals for calculating areas under curves and between curves
• Integration with respect to both $x$ and $y$ (horizontal and vertical slices)

• Volumes of solids of revolution (Disk and Washer methods)
• Applications to physics: Work, force, and Hooke's Law for springs
• Applications to economics: Consumer surplus and producer surplus
• Total accumulation from rate of change (revenue, motion)
• Position functions and rectilinear motion

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Important Formulas

Below are the key formulas used in this exercise:

Area Formulas

$A={\int }_a^{b}f(x)dx\text{(Area under curve, }f(x)\ge 0\text{)}$

$A=-{\int }_a^{b}f(x)dx\text{(Area under curve, }f(x)\le 0\text{)}$

$A={\int }_a^b[f(x)-g(x)]dx\text{(Area between curves, vertical slices)}$

$A={\int }_c^d[f(y)-g(y)]dy\text{(Area between curves, horizontal slices)}$

Key strategy: To find limits of integration when not given, set the two curve equations equal and solve for the intersection points.

Volume Formulas (Solids of Revolution)

$V=\pi {\int }_a^b[f(x){]}^{2}dx\text{(Disk Method — rotation about }x\text{-axis)}$

$V=\pi {\int }_a^b\left([R(x){]}^2-[r(x){]}^2\right)dx\text{(Washer Method)}$

where $R(x)$ is the outer radius and $r(x)$ is the inner radius.

Physics Applications

Hooke's Law: $F=kx$, where $k$ is the spring constant and $x$ is the displacement from natural length.

$W={\int }_a^{b}F(x)dx\text{(Work done by a variable force)}$

$W=\frac{1}{2}k{x}^2\text{(Work to stretch/compress a spring by distance }x\text{)}$

Procedure: Given that a force $F_0$ stretches a spring by $x_0$, find $k=F_0/x_0$, then integrate $F=kx$ over the required interval.

Economic Applications

$\text{Consumer Surplus: }CS={\int }_0^{q^{\ast }}D(q)dq-p^{\ast }{q}^{\ast }$

$\text{Producer Surplus: }PS=p^{\ast }{q}^{\ast }-{\int }_0^{q^{\ast }}S(q)dq$

where $D(q)$ is the demand function, $S(q)$ is the supply function, $p^{\ast }$ is the equilibrium price, and $q^{\ast }$ is the equilibrium quantity.

Accumulation from Rate of Change

$\text{Total quantity}={\int }_a^{b}r(t)dt$

This applies to: total revenue from marginal revenue, total distance from velocity, total population change from growth rate, etc.

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Summary

This exercise comprehensively covers applications of definite integration across geometry, physics, and economics.

• Geometry: Calculate areas between curves using vertical slices ($dx$) or horizontal slices ($dy$); find volumes of solids of revolution using the Disk Method and Washer Method.
• Physics: Calculate work done by variable forces, particularly spring problems using Hooke's Law ($F=kx$).
• Economics: Compute consumer surplus and producer surplus from demand and supply functions.
• Accumulation: Determine total quantities (revenue, displacement, population change) by integrating rate functions.

Key strategies:

1. Find limits of integration from intersection points of curves.
2. Choose the variable of integration ($x$ or $y$) that simplifies the setup.
3. Correctly identify outer and inner radii for the Washer Method.
4. Always verify that the integrand represents the correct quantity (e.g., upper curve minus lower curve for area).