Apply techniques from Unit 4 Topic 1 Sub-topic: Integration techniques to calculate areas between curves determined by functions, with and without technology.

A region is bounded by the curves where \(f(x) = e^{0.5x}\) and \(g(x) = 2\ln(x + 1) + 1\), and the vertical lines \(x = 0\) and \(x = 3\). The table below shows values of both functions at regular intervals over this domain. (a) Using algebraic techniques and the table of values, identify which function forms the upper boundary over the interval. (1 mark) (b) Set up the integral required to calculate the area of the shaded region. (1 mark) (c) Use technology (graphing calculator or computer algebra system) to evaluate the integral and determine the area of the region bounded by the two curves, correct to 2 decimal places. (2 marks)

The region R is bounded by the curves y = sin(2x) and y = cos(x) − 1 for 0 ≤ x ≤ π. (a) Determine the x-coordinates of the points where the curves intersect in the interval [0, π], correct to 2 decimal places. (1 mark) (b) Calculate the area of region R, correct to 2 decimal places. (3 marks)

Calculate the exact area enclosed between the curves $y = \sqrt{x}$ and $y = x^2$ from their point of intersection at the origin to their other point of intersection.