91Ó°ÊÓ

$$ \begin{array}{l}{\text { Find } \int_{3}^{-2} f(x) d x \text { if }} \\\ {\qquad \int_{-2}^{1} f(x) d x=2 \text { and } \int_{1}^{3} f(x) d x=-6}\end{array} $$

Evaluate the integrals using appropriate substitutions. $$ \int \frac{d x}{1+16 x^{2}} $$

True-False Determine whether the statement is true or false. Explain your answer. Each question refers to a particle in rectilinear motion. If the particle has constant nonzero acceleration, its position versus time curve will be a parabola.

Evaluate the integral and check your answer by differentiating. $$ \int\left[\csc ^{2} t-\sec t \tan t\right] d t $$

Determine whether the statement is true or false. Explain your answer. For any continuous function \(f,\) the area between the graph of \(f\) and an interval \([a, b]\) (on which \(f\) is defined) is equal to the absolute value of the net signed area between the graph of \(f\) and the interval \([a, b] .\)

Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. $$ \int_{1 / 2}^{1} \frac{1}{2 x} d x $$

Let \(A\) denote the area between the graph of \(f(x)=\sqrt{x}\) and the interval \([0,1],\) and let \(B\) denote the area between the graph of \(f(x)=x^{2}\) and the interval \([0,1] .\) Explain geometrically why \(A+B=1\)

Find the area under the curve \(y=9 /(x+2)^{2}\) over the interval \([-1,1] .\)

Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. $$ \int_{0}^{1 / \sqrt{2}} \frac{d x}{\sqrt{1-x^{2}}} $$

Evaluate the integrals using appropriate substitutions. $$ \int t \sqrt{7 t^{2}+12} d t $$