1. Let $R$ be the region bounded by the curves $y = x^3$ and $y = x$.

(a) At what points (give the $(x, y)$ coordinates) do the two curves intersect? [6 points]

(a) At what points (give the $(x, y)$ coordinates) do the two curves intersect? [6 points]

(b) Sketch the region. [6 points]

(c) Write down an expression, involving integration, for the area of $R$. [8 points]

2. This problem concerns the definite integral
$\int_0^6 \sqrt{36 - x^2} dx$
(a) Recall that the solution set to the relation $x^2 + y^2 = r^2$ is the circle of radius $r$ centered at the origin. Use this fact to calculate the exact value of the integral. [8 points]

(b) Using 3 approximating rectangles of equal widths and with heights taken at left endpoints, write down and evaluate a Riemann sum to estimate the above integral. [8 points]

(c) Does your answer to (b) underestimate or overestimate the actual integral? Be sure to explain your answer. [4 points]

3. Evaluate the following definite and indefinite integrals. [6 points each]

(a) $\int \frac{4x^4 + \sqrt{x} + 2}{x} dx$

(a) $\int \frac{4x^4 + \sqrt{x} + 2}{x} dx$

(b) $\int_e^{e^{27}} \frac{dx}{x \sqrt[3]{\ln x}}$

(c) $\int \frac{x}{\sqrt{x + 1}} dx$

(d) $\int_{-2}^{2} \sin (2x^3) dx$

4. Suppose $f''(x) = \sec^2 x, f(0) = 2$, and $f'(0) = 0$. [20 points]

(a) Find $f'(x)$.

(a) Find $f'(x)$.

5. Calculate
$\frac{d}{dx} \int_{\sqrt{x}}^{x^2} \sqrt{1 + \sin t^2} dt$

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