Let U be the region in the first quadrant bounded above by
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
Let U be the region in the first quadrant bounded above by y=lnx, below by the x-axis, and on the right by the vertical line x=e3.
(a) Write the definite integral that represents the area of U when vertical slices are used.
(b) Rewrite the integrand from part (a) so that the same area is expressed instead by a single integral with respect to y. Clearly indicate the new limits of integration.
(c) Evaluate both integrals exactly, showing all work. Verify that the numerical results agree.
(d) Explain which antiderivative in parts (a) and (b) can be found with fewer algebraic steps, and justify your choice with reference to standard integration techniques.