4. Найти площадь криволинейной трапеции \( =-1 ; x=1 \) \[ y=x^{3} \quad y=0 ; x=1 ; x=2 \] площадь фигуры, ограниченной графиками функции

4. Let \( g \) be a function such that \( g(1)=3 \) and whose derivative is \( g^{\prime}(x)=6 \cos (2 \pi x)+3 \). Solve analytically a. Use a linear approximation to estimate the value of \( g(1.2) \). b. Find the solution to the differential equation \( d g=(6 \cos (2 \pi x)+3) d x \)

Find equations of the tangent line and normal line to \( x^{2}+4 x y+y^{2}=13,(2,1) \)

5. Given \( h(x)=\int_{0}^{4 x} \sqrt{t-2} d t \), evaluate the following analytically. \( \begin{array}{ll}\text { a. Determine when } h(x)=\frac{16}{3} & \text { b. Determine when } h^{\prime}(x)=8\end{array} \)

23-24 ¿Cómo podría "remover la discontinuidad" en cada una de las siguientes funciones? En otras palabras, ¿cómo redefiniría \( f(2) \) a fin de que sean continuas en \( x=2 \) ? \( \begin{array}{ll}\text { 23. } f(x)=\frac{x^{2}-x-2}{x-2} & \text { 24. } f(x)=\frac{x^{3}-8}{x^{2}-4}\end{array} \)

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.