\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Integrals - Area Under Curves

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Area Under A Curve

To find the area under a curve that is above the x-axis, you just need to integrate the curve between two specific points. This is a natural idea from definite integrals. Here are a couple of videos explaining this.

rootmath - Intro to Area Under a Curve [4min-28secs]

video by rootmath

rootmath - Exact Area Under A Curve [9min-55secs]

video by rootmath

Okay, to really understand this idea, let's work these practice problems.

Practice

Unless otherwise instructed, calculate the area under these curves, between the two points, if given.

\( y=x^2 \); \( [0,4] \)

Problem Statement

Calculate the area under the curve \( y=x^2 \) on the interval \( [0,4] \).

Solution

Michel vanBiezen - 2114 video solution

video by Michel vanBiezen

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\( y=x^3 \); \( [1,3] \)

Problem Statement

Calculate the area under the curve \( y=x^3 \) on the interval \( [1,3] \).

Solution

Michel vanBiezen - 2115 video solution

video by Michel vanBiezen

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\( y=1/x^2 \); \( [1,4] \)

Problem Statement

Calculate the area under the curve \( y=1/x^2 \) on the interval \( [1,4] \).

Solution

Michel vanBiezen - 2117 video solution

video by Michel vanBiezen

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\( y=\sin x \); \( [0,\pi] \)

Problem Statement

Calculate the area under the curve \( y=\sin x \) on the interval \( [0,\pi] \).

Solution

Michel vanBiezen - 2118 video solution

video by Michel vanBiezen

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\(y=x^2-2x+8\); \([1,2]\)

Problem Statement

Calculate the area under the curve \(y=x^2-2x+8\) on the interval \([1,2]\).

Solution

Michel vanBiezen - 2116 video solution

video by Michel vanBiezen

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\( y=\sin x \cos x \); \( [\pi/4, \pi/2] \)

Problem Statement

Calculate the area under the curve \( y=\sin x \cos x \) on the interval \( [\pi/4, \pi/2] \).

Solution

Michel vanBiezen - 2119 video solution

video by Michel vanBiezen

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\(\displaystyle{ y = \frac{x^2}{\sqrt{x^3+9}} }\); \( [-1,1] \)

Problem Statement

Calculate the area under the curve \(\displaystyle{ y = \frac{x^2}{\sqrt{x^3+9}} }\) on the interval \( [-1,1] \).

Solution

Michel vanBiezen - 2120 video solution

video by Michel vanBiezen

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How to Ace Calculus: The Streetwise Guide

When Area Under The Curve is Zero or Negative

Logically, area should always be positive. However, there will be times when you work the problem correctly and you end up with a negative number. What's the deal?

That's easy. When the area is below the x-axis, the area will turn out to be negative. So it is not that we have negative area. The sign tells us where MOST of the area is, above the x-axis for positive area or below the x-axis for negative area.

It is also possible to get zero for the area. This occurs when the area above the x-axis has exactly the same area as the area below the x-axis.

Here is a good video explaining in more detail why the area is negative. He uses an example and explains this very well.

Section 5.3 - Integrals and Negative Area [5min-21secs]

From this discussion, you now understand why knowing what the graphs look like is so important. Let's work these practice problems.

Practice

Unless otherwise instructed, calculate the area under these curves, between the two points, if given. Some of these will have negative or zero area. Make sure you understand why.

\( y = 60x - 6x^2 \); \( 0 \leq x \leq 15 \)

Problem Statement

Calculate the area under the curve \( y = 60x - 6x^2 \) on the interval \( 0 \leq x \leq 15 \). The area may be negative or zero. Make sure you understand why.

Solution

Michel vanBiezen - 2433 video solution

video by Michel vanBiezen

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\(y=x^2-6x\); \( [1,3] \)

Problem Statement

Calculate the area under the curve \(y=x^2-6x\) on the interval \( [1,3] \). The area may be negative or zero. Make sure you understand why.

Final Answer

\(-46/3\)

Problem Statement

Calculate the area under the curve \(y=x^2-6x\) on the interval \( [1,3] \). The area may be negative or zero. Make sure you understand why.

Solution

2434 video solution

Final Answer

\(-46/3\)

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Really UNDERSTAND Calculus

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Topics You Need To Understand For This Page

Related Topics and Links

related topics on other pages

For discussion of area under a parametric curve, see the parametric calculus page.

For discussion of area under a polar curve, see the polar calculus page.

external links you may find helpful

area under curves youtube playlist

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Area Under A Curve - Practice

When Area Under The Curve is Zero or Negative - Practice

Practice Search

Practice Instructions

Unless otherwise instructed, calculate the area under these curves, between the two points, if given.

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