filling the given information, write the appropriate limit of a Riemann sum or definite integrals by filling in the correct information in the parenthesis or bracket. a. $\int_{-1}^{1} f(x)dx = \lim_{n \to \infty} \sum_{k=1}^{n} [\quad ]$; $f(x) = x^2$ b. $\int_{-1}^{1} f(x)dx = \lim_{n \to \infty} \sum_{k=1}^{n} (\quad)[2(-1 + \frac{2k}{n}) - 1]$; $f(x) = 2x - 1$ c. $\lim_{n \to \infty} \sum_{k=1}^{n} [\quad (-1 + \frac{2k}{n})\quad] = \int_{(\quad)}^{(\quad)}(\quad)dx$ d. $\lim_{n \to \infty} \sum_{k=1}^{n} [\quad (\frac{k}{n})^2 - 1\quad] = \int_{(\quad)}^{(\quad)}(\quad)dx$ 6. Suppose that $f$ and $g$ are continuous functions and that $\int_{-1}^{9} f(x)dx = 3$ $\int_{3}^{9} f(x)dx = -2$ $\int_{3}^{9} g(x)dx = 5$ Find each integral. a. $\int_{9}^{3} f(x)dx = \quad$ b. $\int_{3}^{9} [f(x) - g(x)]dx = \quad$ c. $\int_{-1}^{3} f(x)dx = \quad$ d. $\int_{3}^{9} 3g(x)dx = \quad$
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First, let's find the integral of f(x)dx. Given that f(x) = 2x, we can integrate it as follows: ∫f(x)dx = ∫2xdx = x^2 + C, where C is the constant of integration. Show more…
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The Defi nite Integral
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