Solve the initial value v=-1 x=1 Find the volume of the solid of re olution formed by rotating y=sqrt (4-x^3) about the xaxis from x=-2 to x=2 (3 marks) Find the is value Vof the iunction y=2x-x (3 marky) Compute the are length y=ln(xecx) 13 marks Venty the theoren f(x)x^2+2x-8.+1 int _(1)^2(1)/(lnx) Trapezoidal rule (iii) (3 marki)

To solve the initial value problem, we need to find the solution to the differential equation with the initial condition .The general solution to the differential equation is , where is the constant of integration.Using the initial condition , we can substitute into the general solution to find the value of : Therefore, the solution to the initial value problem is .To find the volume of the solid of revolution formed by rotating the region about the x-axis from to , we can use the disk method.The volume is given by the integral: Simplifying the integrand: Integrating term by term: Evaluating the integral: Therefore, the volume of the solid of revolution is .To find the solution to the equation , we can simplify the equation: Therefore, the solution to the equation is .To compute the area under the curve , we can use the definite integral: Using integration by parts, let and . Then and . Simplifying the integral: Evaluating the integral: Therefore, the area under the curve from to is .To evaluate the integral using the trapezoidal rule, we can divide the interval into subintervals of equal width .The trapezoidal rule formula is: In this case, , , and . We need to evaluate at points , where for .Using the trapezoidal rule, we can approximate the integral:$\int_{1}^{2} \frac{1}{\ln(x)} dx \approx \frac{1}{2n} \left[