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Problem 1

Evaluate the integrals. $$ \int_{-1}^{1}\left(x^{2}+2\right) d x $$

Expert verified

The definite integral of the function \(f(x) = x^2 + 2\) over the interval \([-1, 1]\) is \(\frac{14}{3}\).

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Chapter 13

The total cost of producing \(x\) items is given by $$ C(x)=246.76+\int_{0}^{x} 5 t d t $$ Find the fixed cost and the marginal cost of producing the 10th item.

Chapter 13

(Compare Exercise 79 in Section 13.2.) The number of research articles in the prominent journal Physical Review written by researchers in Europe can be approximated by \(E(t)=\frac{7 e^{0.2 t}}{5+e^{0.2 t}}\) thousand articles per year \((t \geq 0)\) where \(t\) is time in years \((t=0\) represents 1983\() .^{49}\) Use a definite integral to estimate the number of articles were written by researchers in Europe from 1983 to 2003 . (Round your answer to the nearest 1,000 articles.) HINT [See Example 7 in Section 13.2.]

Chapter 13

Calculate the total area of the regions described. Do not count area beneath the \(x\) -axis as negative. HINT [See Example 6.] Bounded by the curve \(y=1-x^{2}\), the \(x\) -axis, and the lines \(x=-1\) and \(x=2\)

Chapter 13

Evaluate the integrals. $$ \int_{2}^{3} \frac{x}{2 x^{2}-5} d x $$

Chapter 13

Evaluate the integrals. $$ \int_{0}^{1} x \sqrt{2 x+1} d x $$

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