/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Use geometry to find a formula f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 9

Use geometry to find a formula for \(\int_{0}^{a} x d x,\) in terms of \(a\)

Short Answer

Expert verified
Answer: The formula for the definite integral \(\int_{0}^{a} x dx\) using geometry is \(\frac{1}{2}a^2\).

Step by step solution

01

Sketch the curve and bounded region

Sketch the curve represented by the function \(f(x) = x\) in the interval \([0, a]\). This is a straight line passing through the origin with a slope of 1. Then, shade the area below the curve in the interval \([0, a]\).
02

Identify and analyze the geometrical shape

The area under the curve \(f(x) = x\) within the interval \([0, a]\) creates a triangle. The triangle has a right angle at the origin, its base lies along the x-axis, and its height is determined by the value of \(a\). This means the triangle has vertices: \((0,0)\), \((a,0)\), and \((a,a)\).
03

Calculate the area of the triangle using its dimensions

The area \(A\) of a right triangle can be calculated using the formula: \(A = \frac{1}{2} \text{base} \times \text{height}\). Let the base be \(b\) and the height be \(h\). In this case, we have \(b = a\) along the x-axis and \(h = a\) as the y-coordinate of the vertex \((a,a)\). Therefore, the area of the triangle under the curve is: \(A = \frac{1}{2}a \times a\).
04

Write the definite integral formula in terms of \(a\)

Since the area of the triangle represents the definite integral \(\int_{0}^{a} x dx\), we can now write the formula in terms of \(a\): \[\int_{0}^{a} x dx = \frac{1}{2}a^2\] This gives us the formula for the definite integral \(\int_{0}^{a} x dx\) in terms of \(a\), where the integral represents the area of the triangle under the curve.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=\cos \pi x ; a=0, b=\frac{1}{2}, c=1$$

Evaluate \(\frac{d}{d x} \int_{-x}^{x}\left(t^{2}+t\right) d t\) Separate the integral into two pieces.)

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4}$$

Multiple substitutions Use two or more substitutions to find the following integrals. $$\int_{0}^{\pi / 2} \frac{\cos \theta \sin \theta}{\sqrt{\cos ^{2} \theta+16}} d \theta(\text {Hint}: \text { Begin with } u=\cos \theta .)$$

Use the definition, of the definite integral to justify the property \(\int_{a}^{b} c f(x) d x=c \int_{a}^{b} f(x) d x,\) where \(f\) is continuous and \(c\) is a real number.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.