9. Graphical Applications of Integrals

Find the area of the region enclosed by the inner loop of the curve $r=2+4-\sin \theta$.

Find the area of the region R bounded by the graphs of $y=x^2+1$, $y=2x$, and $x=0$.

Find the area of the region that lies inside both curves: $r=\sin \left(2\theta \right)$ and $r=\cos \left(2\theta \right)$.

Find the area of the region described in the following exercises.

The region bounded by y=e^x, y=2e^−x+1, and x=0

Given the parametric equations $x=t^2-3t$ and $y=t$, for $t\in [0,3]$, find the area enclosed by the curve and the y-axis.

Calculate the area of the shaded region between the 2 functions from $x=0$ to $x=9$

Calculate the area of the shaded region between $f\left(x\right)$ & $g\left(x\right)$ contained between $x=-4$ & $x=-2$.

Sketch the region bounded by $f\left(x\right)=-{(x-2)}^2+5$ & $g\left(x\right)=4x$ on the interval $[0,1]$. Then set up an integral to represent the region's area and evaluate.

Find the area between $f\left(x\right)=x^2-4$ & $g\left(x\right)=-x^2+4$.

Find the area of the shaded region between $f\left(x\right)=x^4-x^2$ & $g\left(x\right)=3x^2$.

Find the area of the shaded region ONLY that lies between the line $y=1$ & $f\left(x\right)=\sin 2x$.

Find the shaded area between $f\left(x\right)=x^3+2x^2$ & $g\left(x\right)=x+2$.

Find the area of the shaded region between $f\left(x\right)=\sin 2x$ & $g\left(x\right)=2\sin x$ from $x=0$ to $x=2\pi$.

Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.

The region bounded by y = 6 cos 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π

Areas of regions Find the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval.

ƒ(𝓍) = 𝓍³ ― 1 on [―1, 2]