9. Graphical Applications of Integrals

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).

137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.

In Exercises 139–142, find the length of each curve.

139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.

In Exercises 139–142, find the length of each curve.

141. y = ln(cos(x)) from x = 0 to x = π/4.

147. Find the area of the region between the curve y = 2x / (1 + x²) and the interval −2 ≤ x ≤ 2 of the x-axis.

Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.

Finding area

Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:

b. π ≤ x ≤ 2π.

Finding area

Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:

c. 2π ≤ x ≤ 3π.

Finding area

Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:

a. π/2 ≤ x ≤ 3π/2.

Finding area

Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:

b. 3π/2 ≤ x ≤ 5π/2.

Consider the region bounded by the graphs of

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

a. Find the area of the region.

Arc length:

Find the length of the curve y = x², 0 ≤ x ≤ √3/2.

7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.

a. lim(x→∞)A(t)

Length of a curve

Find the length of the curve

y = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.