9. Graphical Applications of Integrals

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.

Find the area of each of the regions R₁,R₂, and R₃.

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

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.

Find the area of the region (see figure) in two ways.

a. Using integration with respect to x.

Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on [a, c] (see figure). Assume the curves intersect at x=b.

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = ln x, for 1≤x≤4

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = x³/3, for −1≤x≤1

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = cos 2x, for 0 ≤ x ≤ π

21–30. {Use of Tech} Arc length by calculator

b. If necessary, use technology to evaluate or approximate the integral.

y = cos 2x, for 0 ≤ x ≤ π

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

b. ∫a^b √1+36 cos² 2xdx

35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of

Find the area of the region R.

An area function Consider the functions y = x²/a and y = √x/a, where a>0. Find A(a), the area of the region between the curves.

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

a. ∫a^b √1+16x⁴ dx

123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].

122. Comparing areas The region R₁ is bounded by the graph of y = tan(x) and the x-axis on the interval [0, π/3].

The region R₂ is bounded by the graph of y = sec(x) and the x-axis on the interval [0, π/6]. Which region has the greater area?