Find the area of the region bounded by the curves and lines y=e^x sin(e^x), x=0, y=0, and the curve's first positive intersection with the x-axis.

Question

Find the area of the region bounded by the curves and lines y=e^x sin(e^x), x=0, y=0, and the curve's first positive intersection with the x-axis. I need to know how to do this without a calculator. Please show your work thanks.

Answer 1

f(x) = e^xsin(x)

The first positive crossing is when f(x) = 0.

e^xsin(e^x) = 0

sin(θ) is zero when θ = 0 or π. Use π. So the curve's first x-axis crossing is when e^x = π, or x = ln(π).

The Area will be:

A = ∫₀^ln(π) e^xsin(e^x) dx =

Use u substitution to find the area.

u = e^x, du = e^xdx --> dx = du/e^x

When x = ln(π), u = π

When x = 0, u = 1

∫₁^π sin(u) du = -cos(u)|₁^π = 1 + cos(1) ≈ 1.5403

Well, really need a calculator or trig table to find cos(1), unless you use Newton's approximation to find cos(1)

x_n+1 = x_n + f(x)/f'(x)