You CAN Ace Calculus

 integration definite integrals

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.

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > integrals > area under curves

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

$$y=x^2 ~~ [0,4]$$

Problem Statement

$$y=x^2 ~~ [0,4]$$

Solution

### 2114 solution video

video by Michel vanBiezen

$$y=x^3 ~~ [1,3]$$

Problem Statement

$$y=x^3 ~~ [1,3]$$

Solution

### 2115 solution video

video by Michel vanBiezen

$$y=1/x^2 ~~ [1,4]$$

Problem Statement

$$y=1/x^2 ~~ [1,4]$$

Solution

### 2117 solution video

video by Michel vanBiezen

$$y=\sin x ~~ [0,\pi]$$

Problem Statement

$$y=\sin x ~~ [0,\pi]$$

Solution

### 2118 solution video

video by Michel vanBiezen

$$y=x^2-2x+8 ~~ [1,2]$$

Problem Statement

$$y=x^2-2x+8 ~~ [1,2]$$

Solution

### 2116 solution video

video by Michel vanBiezen

$$y=\sin x \cos x$$ $$[\pi/4, \pi/2]$$

Problem Statement

$$y=\sin x \cos x$$ $$[\pi/4, \pi/2]$$

Solution

### 2119 solution video

video by Michel vanBiezen

$$\displaystyle{ y = \frac{x^2}{\sqrt{x^3+9}} ~~ [-1,1] }$$

Problem Statement

$$\displaystyle{ y = \frac{x^2}{\sqrt{x^3+9}} ~~ [-1,1] }$$

Solution

### 2120 solution video

video by Michel vanBiezen

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. Some of these will have negative or zero area. Make sure you understand why.

Calculate the area under the curve $$y = 60x - 6x^2$$ on $$0 \leq x \leq 15$$.

Problem Statement

Calculate the area under the curve $$y = 60x - 6x^2$$ on $$0 \leq x \leq 15$$.

0

Problem Statement

Calculate the area under the curve $$y = 60x - 6x^2$$ on $$0 \leq x \leq 15$$.

Solution

### 2433 solution video

video by Michel vanBiezen

0

Calculate the area under the curve $$y=x^2-6x$$ between 1 and 3.

Problem Statement

Calculate the area under the curve $$y=x^2-6x$$ between 1 and 3.

$$-46/3$$

Problem Statement

Calculate the area under the curve $$y=x^2-6x$$ between 1 and 3.

Solution

### 2434 solution video

$$-46/3$$