9. Graphical Applications of Integrals

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)

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.)