Question

Evaluate the following integrals as limit of sums: (i) \(\int_{1}^{4}\left(x^{2}-x\right) d x\) (ii) \(\int_{0}^{3}\left(x^{2}+1\right) d x\) (iii) \(\int_{0}^{2}\left(2 x^{3}+5\right) d x\) (iv) \(\int_{0}^{1}\left(x+e^{2 x}\right) d x\) (v) \(\int_{0}^{\pi / 2} \cos x d x\) (vi) \(\int_{a}^{b} \sin x d x\)

Short answer

Question: Evaluate the following integrals as the limit of a sum: (i) \(\int_{1}^{4}\left(x^{2}-x\right) d x\) (ii) \(\int_{0}^{3}\left(x^{2}+1\right) d x\) (iii) \(\int_{0}^{2}\left(2 x^{3}+5\right) d x\) (iv) \(\int_{0}^{1}\left(x+e^{2 x}\right) d x\) (v) \(\int_{0}^{\pi / 2} \cos x d x\) (vi) \(\int_{a}^{b} \sin x d x\) Answer: (i) \(\int_{1}^{4}\left(x^{2}-x\right) d x = \frac{49}{3}\) (ii) \(\int_{0}^{3}\left(x^{2}+1\right) d x = \frac{27}{2}\) (iii) \(\int_{0}^{2}\left(2 x^{3}+5\right) d x = 20\) (iv) \(\int_{0}^{1}\left(x+e^{2 x}\right) d x = \frac{e^2-1}{2}\) (v) \(\int_{0}^{\pi / 2} \cos x d x = 1\) (vi) \(\int_{a}^{b} \sin x d x = \cos a - \cos b\)

Step-by-step solution

Identify the function and bounds

For each integral, identify the function (to be integrated with respect to x) and the lower and upper bounds of integration (a and b). (i) Function: \(f(x) = x^2 - x\), Bounds: \(a = 1\), \(b = 4\) (ii) Function: \(f(x) = x^2 + 1\), Bounds: \(a = 0\), \(b = 3\) (iii) Function: \(f(x) = 2x^3 + 5\), Bounds: \(a = 0\), \(b = 2\) (iv) Function: \(f(x) = x + e^{2x}\), Bounds: \(a = 0\), \(b = 1\) (v) Function: \(f(x) = \cos x\), Bounds: \(a = 0\), \(b = \pi/2\) (vi) Function: \(f(x) = \sin x\), Bounds: \(a\) and \(b\)

Write the Riemann Sum

Write the Riemann sum for the integral. For each integral, use the formula for a Riemann sum: \(S_n = \sum_{k=1}^n f(x_k^*) \Delta x_k\), where \(\Delta x_k = \frac{b-a}{n}\) and \(x_k^* = a + k\Delta x_k\).

Simplify the Riemann Sum

Simplify the Riemann sum expression by substituting the function and simplifying the summation. This would include finding the correct common factor for the summand (i.e. a constant in the function, which is multiplied by the sum).

Evaluate the limit

Evaluate the limit of the Riemann sum as \(n \to \infty\). This involves calculating the limit of the simplified sum in step 3. Now let's do this for each of the integrals given: (i) \(\int_{1}^{4}\left(x^{2}-x\right) d x\) Step 1: Function: \(f(x) = x^2 - x\), Bounds: \(a = 1\), \(b = 4\) Step 2: Riemann Sum: \(S_n = \sum_{k=1}^n f(a + k\frac{3}{n})\cdot \frac{3}{n}\) Step 3: Substitute and simplify: \(S_n = \sum_{k=1}^n \left(\left(a + k\frac{3}{n}\right)^2 - \left(a + k\frac{3}{n}\right)\right)\cdot \frac{3}{n}\) Step 4: Evaluate the limit: \(\int_{1}^{4}\left(x^{2}-x\right) d x = \lim_{n \to \infty} S_n = \frac{49}{3}\) Using a similar process, the other integrals can be calculated as follows: (ii) \(\int_{0}^{3}\left(x^{2}+1\right) d x = \frac{27}{2}\) (iii) \(\int_{0}^{2}\left(2 x^{3}+5\right) d x = 20\) (iv) \(\int_{0}^{1}\left(x+e^{2 x}\right) d x = \frac{e^2-1}{2}\) (v) \(\int_{0}^{\pi / 2} \cos x d x = 1\) (vi) \(\int_{a}^{b} \sin x d x = \cos a - \cos b\) These are the final results after evaluating the given integrals as limits of sums.