find, correct to three decimal places, the area of the region bounded by the curve y=e^x, y=e^3x, x=1?

Answer 1

First, you need to know where the two equations equal each other so that you have the third intersection point of the three functions/lines that are the bounds of the area. (The first two intersection points, of course, are where e^x and e^3x intersect with x=1.) e^x = e^3x where x=3x, so x=0. So, are bounds are from x=0 to x=1.

To find the area, take the integral of the greater equation (e^3x) and subtract the integral of the lesser equation (e^x). When working with integrals of exponent functions, know that the integral of e^Ax = (1/A)*e^Ax, where A is some constant. So, the integral of e^3x is (1/3)*e^3x. Evaluate that from 0 to 1. The integral of e^x is just e^x. Evaluate that from 0 to 1, and subtract that value from the first evaluate integral. That's your answer.

Answer 2

You did no longer say locate the section, although that's implied by ability of the answer you wrote. the answer would desire to truly be in terms of y: ?(e^Y +a million) dY, from y = 0 to 3, although, you will get the comparable answer in case you have x as a replace of y, that's in basic terms a letter. The 0 and 3 are the barriers alongside the y-axis. This bounded area spans from y = 0 as much as the horizontal boundary line y=3. i'm no longer prepared on writing the respond in terms of x (whether it produces the comparable numerical crucial), because of the fact it loses the meaning of integrating w.r.t. the y-axis.

Answer 3

You will need a integral. Graph the curves to figure out the bounds of the integral.