9. Graphical Applications of Integrals

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)

Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

a. Find an expression for L.

Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

b. Is L concave up or concave down on [0, ∞)?

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .

(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2. 

Area functions for the same linear function Let ƒ(t) = 2t ― 2 and consider the two area functions A (𝓍) = ∫₁ˣ ƒ(t) dt and F(𝓍) = ∫₄ˣ ƒ(t) dt .

(a) Evaluate A (2) and A (3). Then use geometry to find an expression for A (𝓍) , for 𝓍 ≥ 1 .

Determine the area of the shaded region in the following figures.

Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)

Determine the area of the shaded region in the following figures.

Determine the area of the shaded region in the following figures.

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

The region bounded by y=4x+4, y=6x+6, and x=4

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

The region bounded by y=e^x, y=e^−2x, and x=ln 4

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

The region bounded by y=2 / 1 + x^2 and y=1

14–25. {Use of Tech} Areas of regions Determine the area of the given region.

The region bounded by y = ln x,y = 1, and x = 1

14–25. {Use of Tech} Areas of regions Determine the area of the given region.

The region bounded by y = x²,y = 2x²−4x, and y = 0

14–25. {Use of Tech} Areas of regions Determine the area of the given region.

The region in the first quadrant bounded by y = x/6 and y = 1−|x/2−1|