Given that \( \int_{0}^{1}\left(2+3 \cos \frac{1}{2} x\right) d x=\int_{-1}^{2} e^{2 x+1} d x \), find the value of \( k \), giving your answer correct to 3 decimal places.

\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. \\ 1.) \( f(x)=x^{2}-3 x+4,[1,4], n=6 \) \\ a) Left Rectangles \\ b) Right Rectangles \\ d) Trapezoids \\ c) Midpoint Rectangles \\ (n) \\ \\ \hline\end{tabular}

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

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)

- \( \int x \cos x d x=u \cdot v-\int v d u \)

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