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Calculus: Integration
Integration is a primary topic in most second semester calculus
courses. There are four applications of integration that can be
found below -- finding areas (as in
Ellipse Areas),
finding volumes
(Cup Capacity),
differential equations
(Vehicle Value),
and arc length
(Painting the Sin Curve).
For background information elsewhere on our site, explore the
High School Calculus
area of the Ask Dr. Math archives. For example, the archives provide
information on integrals and the fundamental
theorem of calculus. To find relevant sites on the Web, browse and search
Calculus (Single Variable):
Integration
in our Internet Mathematics Library.
Access to these problems requires a Membership.
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Be My Valentine
- Leigh Nataro
-
Trig/Calc, difficulty level 4. Find the areas of two heart-shaped valentines.
... more>>
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Cooling Colas
- Leigh Nataro
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Trig/Calc, difficulty level 3. Use Newton's Law of cooling to determine when the colas will freeze.
... more>>
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Cup Capacity
- Leigh Nataro
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Trig/Calc, difficulty level 3. Find the volume of a cup based on its dimensions.
... more>>
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Cutting the Cookie
- Leigh Nataro
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Trig/Calc, difficulty level 2. Where should you cut a circular cookie so its area is in a ratio of 1:2 ?
... more>>
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Doughnut Dilemma
- Leigh Nataro
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Trig/Calc, difficulty level 2. Which doughnut shop makes a doughnut with a greater volume?
... more>>
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An Eggstra Special Problem
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the area of a cross-section of an egg given some dimensions of the cross-section.
... more>>
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Ellipse Areas
- Leigh Nataro
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Trig/Calc, difficulty level 3. Use calculus to show that the area of an ellipse is pi*a*b.
... more>>
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Find the Function
- Leigh Nataro
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Trig/Calc, difficulty level 3. Find a cubic polynomial that has a given maximum, minimum, and point of inflection.
... more>>
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Five Golden Rings
- Leigh Nataro
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Trig/Calc, difficulty level 5. Based on the given information, find the volume of five golden rings.
... more>>
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Football Volume
- Leigh Nataro
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Trig/Calc, difficulty level 3. Given the dimensions of a football, find its volume.
... more>>
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Googol the Giant's Gargantuan Goblet
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the volume of Googol the Giant's elliptical paraboloid goblet.
... more>>
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My Hot Chocolate is Too Hot!
- Leigh Nataro
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Trig/Calc, difficulty level 3. Use Newton's Law of Cooling to determine how long it will take for your hot chocolate to cool.
... more>>
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Needles and Lines
- Leigh Nataro
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Trig/Calc, difficulty level 5. Use trigonometry and calculus to find the probability of a needle touching a line.
... more>>
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One Area - Many Solutions
- Leigh Nataro
-
Trig/Calc, difficulty level 2. Find the area of a triangle given the coordinates of its vertices.
... more>>
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The Oval Braided Rug
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the area of a shaded portion of an ellipse.
... more>>
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Painting the Sine Curve
- Leigh Nataro
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Trig/Calc, difficulty level 4. Find the length of the sine curve that would fill a 6-ft by 100-ft rectangle.
... more>>
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A Parabola Proof
- Leigh Nataro
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Trig/Calc, difficulty level 3. Prove that the area of a parabola is 2/3 the product of its width and height.
... more>>
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A Picture Is Worth A Thousand Words
- Leigh Nataro
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Trig/Calc, difficulty level 4. Find the ratio of the area of the overlapping region to the area of non-overlapping regions of two circles.
... more>>
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The Square and the Line
- Leigh Nataro
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Trig/Calc, difficulty level 1. Find the equation of the line that will split a unit square into two pieces that have equal areas.
... more>>
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The Suspension Bridge
- Leigh Nataro
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Trig/Calc, difficulty level 4. Determine the length of a parabola-shaped cable on a suspension bridge.
... more>>
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That's A Lot of Rock!
- Leigh Nataro
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Trig/Calc, difficulty level 3. Find the volume of a tunnel through a mountain.
... more>>
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Time to Make the Doughnuts
- David Turner
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Trig/Calc, difficulty level 4. Find the radius of a doughnut with center at (0, 10) and a volume of 3158 cubic centimeters.
... more>>
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Vehicle Value
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the future value of a car based on its current price and initial selling price.
... more>>
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