9. Graphical Applications of Integrals

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.

∫₀ᵃ ƒ(𝓍) d𝓍

What is the area of the region bounded by the lines $x=-5$, $x=1$, and the curves $y=10x$ and $y=x^2-11$?

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

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

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = (1/x) over [c, c + 1]

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.

Find the area of the region enclosed by the inner loop of the curve $r=6+12 \sin \left(\theta \right)$.

A family of exponential functions

b. Verify that the arc length of the curve y=f(x) on the interval [0, ln 2] is A(2^a-1) - 1/4a²A (2^-a - 1).

{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of ƒ and the 𝓍-axis on the given interval. Graph ƒ and show the region 𝑅.                                              

 ƒ(𝓍) = 2 ― |𝓍| on [ ― 2 , 4]

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

y = sin x, y = x, 0 ≤ x ≤ π/4

Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.

Finding arc length

Find the length of the curve

y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

Determine whether the following statements are true and give an explanation or counterexample.

b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

Let R be the region bounded by the graphs of $y=2x$ and $y=4x-x^2$. What is the area of R?

Consider the region bounded by the graphs of

y = ln(x), y = 0, and x = e.

a. Find the area of the region.