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Differential Equations and Riemann Sums

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Solve the differential equation:

1. f"(x) = x^2, f(0) = 0, f'(0) = 1

2. Find the area bounded by the graph of f(x) = x*sqrt(1 - x^2), the x-axis and the vertical lines x = 0 and x = 1.

3. Given the function f(x) = x^3 defined for 0 < x < 1 evaluate the lower and upper Riemann sums.

4. Use the fundamental theorem of calculus to evaluate the following definite integrals:

a) the integral of |x^2 - 4| dx
b) the integral of | sin(x) - cos(x)| dx

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Differential equations and Riemann sums are solved.

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