The purpose of this problem is to compute the area of the region R bounded by a curve of the form y = cx2/3 , x ≥ 0, the tangent line to that curve at the point P with coordinates (a, b) in the first quadrant, and the negative x-axis. (a) Parameterize the curve with an equation of the form y = cx^2/3 , x ≥ 0, that passes through the point P in the first quadrant with coordinates (a, b). Choose a parameterization of the form x = f1(t) = kt^p y = g1(t) = mt^q , where k, m are constants that depend on a and b (but not c), and p, q are postive integers. Include a domain for your parameterization and verify that your parameterization is correct. (b) Suppose the coordinates x and y are measured in meters. What are the units of the parameter t in part (a)? Explain your reasoning. (c) Use the parameterization of the curve from part (a) to parameterize the tangent line to the curve at P. Be sure to include a domain for the parameterization. Do this by first using the parameterization of the curve to find a Cartesian equation of the tangent line at P. Then using the function y = g1(t) from part (a) for the y-coordinate function of the line, find an expression for the x-coordinate function.

The purpose of this problem is to compute the area of the region R bounded by a curve of the form y = cx2/3 , x ≥ 0, the tangent line to that curve at the point P with coordinates (a, b) in the first quadrant, and the negative x-axis. (a) Parameterize the curve with an equation of the form y = cx^2/3 , x ≥ 0, that passes through the point P in the first quadrant with coordinates (a, b). Choose a parameterization of the form x = f1(t) = kt^p y = g1(t) = mt^q , where k, m are constants that depend on a and b (but not c), and p, q are postive integers. Include a domain for your parameterization and verify that your parameterization is correct. (b) Suppose the coordinates x and y are measured in meters. What are the units of the parameter t in part (a)? Explain your reasoning. (c) Use the parameterization of the curve from part (a) to parameterize the tangent line to the curve at P. Be sure to include a domain for the parameterization. Do this by first using the parameterization of the curve to find a Cartesian equation of the tangent line at P. Then using the function y = g1(t) from part (a) for the y-coordinate function of the line, find an expression for the x-coordinate function. (d) Use your parameterizations from parts (a) and (c) to find simplified ex- pressions (all in terms of only t and/or dt) for the following: (i) the length of a differential rectangle (parallel to the x-axis) of the region R described above in terms of the parameter t. Check that your expression is dimensionally correct. Include a picture showing the region, the differential rectangle, and the coordinates of the endpoints of the differential rectangle to help with your explanation. (ii) the width of the differential rectangle. Check that your expression is dimensionally correct. (iii) the area of the differential rectangle. Check that your expression is dimensionally correct. (e) Use part (d) to set up a definite integral that gives the area of the re- gion. Explain the role of the integral and how you determined the limits of integration. (f ) Evaluate the definite integral from part (e) to find a simplified expres- sion for the area of the region. Check that your expression is dimensionally correct. (g) Compare the area in part (f ) to the area of ∆OP Q, where O is the origin and Q the x-intercept of the tangent line to the curve at P. What fraction of the area of the triangle does the region R occupy?