2. Antiderivative for ln(x) Recall that [Ima In x dx = x ln x - x...

Transcribed Image Text:

2. Antiderivative for ln(x) Recall that [Ima In x dx = x ln x - x + C. We first determined this antiderivative using integration by parts. In this problem, you will give a geometric explanation why this is the case (avoiding integration by parts). The purpose of the problem is not to figure out the formula (we know it now), but to show another visual way to see why it is correct. Let a be a constant which could be any real number. (We call this value a a parameter.) (a) Consider the area between the graph of eª and the x-axis from x = 0 to x = a using vertical slices. Write down an integral expression for this area and compute it. = (b) Now consider the area between the graph of eª and the y-axis from y 1 to y = eº using horizontal slices. Write down an integral expression for this area, from horizontal slicing, but do not compute it. Use t for the variable of integration. (c) Draw a picture a picture of these areas on the same set of axes. (For specificity, you may want to let a = 1.) Using geometric reasoning, what is the sum of the two areas in parts (a) and (b)? (d) Show how you can conclude from the previous part that (e) How can we conclude that using this last part? S 1 In t dt = x lnxx +1. [in. In x dx = x ln x - x + C. 2. Antiderivative for ln(x) Recall that [Ima In x dx = x ln x - x + C. We first determined this antiderivative using integration by parts. In this problem, you will give a geometric explanation why this is the case (avoiding integration by parts). The purpose of the problem is not to figure out the formula (we know it now), but to show another visual way to see why it is correct. Let a be a constant which could be any real number. (We call this value a a parameter.) (a) Consider the area between the graph of eª and the x-axis from x = 0 to x = a using vertical slices. Write down an integral expression for this area and compute it. = (b) Now consider the area between the graph of eª and the y-axis from y 1 to y = eº using horizontal slices. Write down an integral expression for this area, from horizontal slicing, but do not compute it. Use t for the variable of integration. (c) Draw a picture a picture of these areas on the same set of axes. (For specificity, you may want to let a = 1.) Using geometric reasoning, what is the sum of the two areas in parts (a) and (b)? (d) Show how you can conclude from the previous part that (e) How can we conclude that using this last part? S 1 In t dt = x lnxx +1. [in. In x dx = x ln x - x + C.