Area by Integration

Smallest Part From The Circle That Was Divided Into Four Parts By Perpendicular Chords

Problem
Divide the circle of radius 13 cm into four parts by two perpendicular chords, both 5 cm from the center. What is the area of the smallest part.
 

03 Area Enclosed by Cardioids: r = a(1 + sin θ); r = a(1 - sin θ), r = a(1 + cos θ), r = a(1 - cos θ)

Problem
Find the area individually enclosed by the following Cardioids:
(A)   $r = a(1 - \cos \theta)$
(B)   $r = a(1 + \cos \theta)$
(C)   $r = a(1 - \sin \theta)$
(D)   $r = a(1 + \sin \theta)$
 

003-cardioid-neg-pos-sine-cosine.gif

 

08 Area Enclosed by r = a sin 3θ and r = a cos 3θ

Problem
Find the area bounded by $r = a \sin 3\theta$ and $r = a \cos 3\theta$.
 

008-polar-area-three-leaf_rose_sine_cosine.gif

 

05 Area Enclosed by r = a sin 2θ and r = a cos 2θ

Problem
Find the area bounded by $r = a \sin 2\theta$ and $r = a \cos 2\theta$.
 

008-polar-area-four-leaf_sine_cosine.gif

 

01 Area Enclosed by r = 2a cos^2 θ

Problem
Find the area enclosed by r = 2a cos2 θ.
 

004-polar-area-two-leaf-rose-integration.gif

 

07 Area Enclosed by r = 2a cos θ and r = 2a sin θ

Problem
Find the area enclosed by the following:

(a)   $r = 2a \cos \theta$
(b)   $r = 2a \sin \theta$

 

001-polar-area-circle_01.gif

 

Example 8 | Area bounded by arcs of quarter circles

Problem
Arcs of quarter circles are drawn inside the square. The center of each circle is at each corner of the square. If the radius of each arc is equal to 20 cm and the sides of the square are also 20 cm. Find the area common to the four circular quadrants. See figure below.
 

018-arcs-of-quarter-circles-common-area.gif

 

Area for grazing by the goat tied to a silo

Problem
A goat is tied outside a silo of radius 10 m by a rope just long enough for the goat to reach the opposite side of the silo. Find the area available for grazing by the goat. Note that the goat may not enter the silo.
 

grazing-area-goat-figure-1.jpg

 

Cycloid: equation, length of arc, area

Problem
A circle of radius r rolls along a horizontal line without skidding.

  1. Find the equation traced by a point on the circumference of the circle.
  2. Determine the length of one arc of the curve.
  3. Calculate the area bounded by one arc of the curve and the horizontal line.
cycloid_small_02.gif

 

709 Centroid of the area bounded by one arc of sine curve and the x-axis

Problem 709
Locate the centroid of the area bounded by the x-axis and the sine curve $y = a \sin \dfrac{\pi x}{L}$ from x = 0 to x = L.
 

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