Tutorial Week 11
Computing Integrals
Q1: Compute \(\int x^3 + 2x + 2\, dx\).
Q2: Compute \(\int \sqrt[6]{x^3} \, dx\).
Q3: Compute \(\int \frac{1}{2}x + \frac{5}{x^2 + 1} \, dx\).
Q4: Compute \(\int \frac{t^5 + t^4}{t} + cos(t) + sin(t) \, dt\).
Q5: Compute \(\int_{0}^{1} 5x + x^2\sqrt[3]{x} \, dx\).
Riemann Sums
Q6: Write \(\int_{3}^{9} (x^4 + 2x) \, dx\) as the limit of a Riemann sum.
Q7: Write \(\int_{-\pi}^{2\pi} sin(x) \, dx\) as the limit of a Riemann sum.
Q8: Evaluate \(\int_{0}^{2} {5x + 1} \, dx\) using Riemann sums.
Approximating Integrals using Riemann Sums
Q9: Approximate \(\int_{0}^{1} 10 - x^2 \, dx\) using the right Riemann sum over four rectangles.
Q10: If \(f''(x) = x + 2 + cos(x)\), \(f(0) = 1\), and \(f(\pi) = 2\), find \(f(x)\).
