2 Applications of Integration Free Online FlashCards

How is area between curves computed using vertical slices?

For functions of x, subtract the lower graph from the upper graph: $A=\int_a^b[f(x)-g(x)]\,dx$. If horizontal slices are simpler, use right minus left with respect to $y$.

What is the general slicing formula for volume?

The slicing method adds cross-sectional areas: $V=\int_a^b A(x)\,dx$, where $A(x)$ is the area of a slice perpendicular to the x-axis.

What formula gives volume by washers?

The washer method subtracts the inner disk from the outer disk: $V=\pi\int_a^b\left([R(x)]^2-[r(x)]^2\right)\,dx$.

When is the cylindrical-shell method especially convenient?

For rotation around the y-axis, cylindrical shells give $V=2\pi\int_a^b x f(x)\,dx$ when x is the radius and $f(x)$ is the height.

What is the arc-length formula for $y=f(x)$?

For $y=f(x)$, the arc length is $L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx$.

What differential quantity underlies surface area of revolution?

Surface area is circumference times differential arc length: $dS=2\pi(\text{radius})\,ds$. The radius must be a nonnegative distance from the axis.

How do you calculate a function’s average value on $[a,b]$?

The average value is total accumulation divided by interval length: $f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$.

What integral represents work done by a variable force?

For a variable force parallel to displacement, work is $W=\int_a^b F(x)\,dx$. A force opposing displacement contributes negative work.

What work is required to move a spring from $x=a$ to $x=b$?

For a spring with Hooke’s law $F(x)=kx$, the work from $x=a$ to $x=b$ is $W=\frac{k}{2}(b^2-a^2)$.

What factors appear in a pumping-work integral?

Pumping work integrates weight times lifting distance: $W=\int_a^b\delta A(y)D(y)\,dy$, where $D(y)$ measures the distance to the destination.

How is hydrostatic pressure related to depth?

At depth $s$ in a fluid of weight density $\delta$, pressure is $p=\delta s$. Pressure increases with depth below the fluid surface.

What is the hydrostatic-force formula for a vertical plate?

For a vertical plate, hydrostatic force is $F=\int_a^b\delta s(y)w(y)\,dy$, integrating pressure times the strip’s width and thickness.