\( - \varepsilon < \frac { 1 } { x ^ { 2 } + 1 } - 1 < 0 \)

Given that \( \int_{1}^{4} f(x) \mathrm{d} x=5 \), evaluate \( \begin{array}{ll}\text { (a) } \int_{4}^{1} 2 f(x) \mathrm{d} x, & \text { (b) } \int_{1}^{3} \mathrm{f}(x) \mathrm{d} x+\int_{3}^{4}[f(x)+3] \mathrm{d} x\end{array} \)

Evaluate the integral. \( \int_{0}^{1} \sin (3 \pi t) d t \)

For each problem, approximate the area under the given function and above the \( x \)-axis on the given interval using the specified number of rectangles/trapezoids. You may use your calculator. 1.) \( f(x)=x^{2}-3 x+4,[1,4], n=6 \) a) Left Rectangles b) Right Rectangles c) Midpoint Rectangles d) Trapezoids (a)

\begin{tabular}{l} For each problem, approximate the area under the given function and above the \( x \)-axis on the given \\ interval using the specified number of rectangles/trapezoids. You may use your calculator. \\ a) Left Rectangles \\ 1.) \( x^{2}-3 x+4,[1,4], n=6 \) \\ b) Right Rectangles \\ d) Trapezoids \\ \hline\end{tabular}

For each problem, approximate the area under the given function and above the \( x \)-axis on the give interval using the specified number of rectangles/trapezoids. You may use your calculator. 1.) \( f(x)=x^{2}-3 x+4,[1,4], n=6 \) a) Left Rectangles b) Right Rectangles c) Midpoint Rectangles d) Trapezoids