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6:13{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.
The right Riemann sum for Ζ(π)) = x + 1 on [0, 4] with n = 50.
Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.
The region bounded by y = 6 cos π and the π-axis between π = βΟ/2 and π = Ο
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
β«α΅ββ Ζ(g(π)) dπ
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βαΆ |Ζ(π)| dπ
Derivatives of integrals Simplify the following expressions.
d/dπ β«βΛ£ (β1 + tΒ²) dt (Hint: β«Λ£ββ (β1 + tΒ²) dt = β«β°ββ (β1 + tΒ²) dt + β«Λ£ββ (β1 + tΒ²) dt ) .
Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length βπ? List the grid points xβ , xβ , xβ , xβ and xβ. Which points are used for the left, right, and midpoint Riemann sums?
