Question

Definite IntegralsIn Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.     n lim  ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0] ∥P∥→0  k = 1

Accepted Answer

Recognize that the given limit of the sum is a Riemann sum representing a definite integral over the interval $$[-\pi, 0]. $$The sum is of the form $$\sum_{k=1}^n f(c_k) \Delta x_k$$, where $$f(x) = \cos\left(\frac{x}{2}ight). $$Identify the interval of integration from the partition $$P$$, which is $$[-\pi, 0]. $$The limit as $$\|P\| 	o 0$$ means the norm of the partition approaches zero, ensuring the sum approaches the integral. Express the limit as the definite integral:

$$\lim_{\|P\| 	o 0} \sum_{k=1}^n \cos\left(\frac{c_k}{2}ight) \Delta x_k = \int_{-\pi}^0 \cos\left(\frac{x}{2}ight) \, dx To $$evaluate the integral, use a substitution method. Let $$u = \frac{x}{2}$$, so that $$dx = 2 \, du. $$Adjust the limits of integration accordingly: when $$x = -\pi$$, $$u = -\frac{\pi}{2}$$; when $$x = 0$$, $$u = 0. $$Rewrite the integral in terms of $$u$$ and integrate:

$$\int_{-\frac{\pi}{2}}^0 \cos(u) \cdot 2 \, du = 2 \int_{-\frac{\pi}{2}}^0 \cos(u) \, du$$

Then compute the integral of $$\cos(u)$$ over the new limits.

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Key Concepts

Definite Integral as a Limit of Riemann Sums

Partition and Norm of a Partition

Evaluating Definite Integrals of Trigonometric Functions