9. Graphical Applications of Integrals

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]

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) = sin 𝓍 on [―π/4, 3π/4]

Use geometry and properties of integrals to evaluate the following definite integrals.                                                                                          

 ∫₀⁴ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.                                              

 ƒ(𝓍) = 16―𝓍² on [―4, 4]

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.                                              

 ƒ(𝓍) = 2 sin 𝓍/4 on [0, 2π]

Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 𝓍⁴ ― 𝓍² on [―1, 1]

41–48. Geometry problems Use a table of integrals to solve the following problems.

43. Find the length of the curve y = eˣ on the interval from 0 to ln 2.

Let ƒ(𝓍) = c, where c is a positive constant. Explain why an area function of ƒ is an increasing function.

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(a) A(2)

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(8)