The area bound by the graph of f(x)=ex−1 f(x)=e^x-1 , the x-axis and the line x=2 is equal to:

Question

BlackTom AI · Accepted Answer

We start by identifying the region whose area is asked for: it lies under the curve f(x) = e^x - 1, above the x-axis, and limited on the right by the vertical line x = 2. First, find where the curve intersects the x-axis to set the lower limit of integration: e^x - 1 = 0 implies e^x = 1, so x = 0. Therefore, the area is the integral from 0 to 2 of (e^x - 1) dx.

Option a: e^2 - 2. This would correspond to integrating e^x alone from 0 to 2, giving e^2 - 1, and subtracting 1 gives e^2 - 2 only if you misaccounted the constant, but the correct evaluation is e^2 - 3, so this is incorrect.

Option b: e^2 + 1. This would arise from an incorrect sign or incorrect lower limit, since the actual integral yields e^2 - 3, not e^2 + 1. The presence of +1 instead of -3 indicates a miscalculation of the constant term.

Option c: e^2 + 2. Similar to option b, this option incorrectly adds a positive constant that does not come from the definite integral ∫0^2 (e^x - 1) dx, where the constant term should be negative due to the -1 in the integrand.

Option d: e^2 - 3. This matches the correct evaluation. Compute step by step: ∫0^2 (e^x - 1) dx = [e^x]0^2 - [x]0^2 = (e^2 - 1) - (2 - 0) = e^2 - 1 - 2 = e^2 - 3. The result e^2 - 3 is the exact area under the curve from x = 0 to x = 2, above the x-axis, bounded by x = 2.

In summary, the correct area is obtained by integrating e^x - 1 from 0 to 2, yielding e^2 - 3, so option d is the correct choice. The other options reflect common mistakes such as omitting the lower limit contribution, mismanaging the constant term from the -1, or miscomputing the definite integral.

MUF0092 Mathematics Unit 2 - Semester 2, 2025 3.1 Integration checking understanding

The area bound by the graph of f(x)=ex−1 f(x)=e^x-1 , the x-axis and the line x=2 is equal to:

查看解析

登录即可查看完整答案

类似问题

Find the signed area produced by:[math: A=∫364cos(5x)dx](square units)Calculate your answer correct to 4 decimal places.(Note: Make sure your calculator is in radian mode.)

Find the area produced by:[math: A=∫16(8x3+12x2+4)dx](square units)

Evaluate the integral.

Evaluate the integral.

Graph the integrand and use geometry to evaluate the integral.

Find the area of the shaded region.

Graph the integrand and use geometry to evaluate the integral.

Solve the problem. Suppose that = 3. Find and .

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单