Show π/6 ≤ ∫₀^(π/2) sin x / (1+x²) dx ≤ π/2.
\subsection*Solution 7 This is the standard definition of the Riemann integral using right endpoints. Since (f) is continuous, it is Riemann integrable, and the limit of any sequence of Riemann sums with mesh (\to 0) equals the integral.
# Riemann Integral: Problems and Solutions Problem 1 Compute the Riemann sum for f(x) = x² on [0,2] using 4 subintervals and right endpoints. riemann integral problems and solutions pdf
Let u = x², du = 2x dx → (1/2)∫₀¹ e^u du = (e-1)/2.
\subsection*Solution 1 [ \Delta x = \frac2-04 = 0.5,\quad x_i^* = 0.5,1,1.5,2. ] [ S = \sum_i=1^4 f(x_i^*)\Delta x = (0.25+1+2.25+4)\times0.5 = 7.5\times0.5 = 3.75. ] Show π/6 ≤ ∫₀^(π/2) sin x / (1+x²) dx ≤ π/2
\beginenumerate[label=\arabic*.] \item (\int_0^1 (3x^2-2x+1)dx = 1) \item (\int_1^e \frac1xdx = 1) \item (\int_0^\pi/2 \sin 2x,dx = 1) \item (\int_0^4 |x-2|dx = 4) \item (\lim_n\to\infty \sum_k=1^n \fracnn^2+k^2 = \frac\pi4) \endenumerate
No. Upper sum = 1, lower sum = 0 for any partition, so inf U ≠ sup L. Intermediate Problems Problem 4 ∫₀¹ x e^(x²) dx. # Riemann Integral: Problems and Solutions Problem 1
\subsection*Problem 5 Use the comparison property of the Riemann integral to show: [ \frac\pi6 \le \int_0^\pi/2 \frac\sin x1+x^2,dx \le \frac\pi2. ]