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Definite integral is calculated as:
Here is the indefinite integral of :
Evaluate each of the following integrals:
(a)
(b)
(c)
Solutions:
(a) ;
(b) ;
(c)
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Evaluate each of the following integrals:
(a)
(b)
(c)
Solutions:
(a) ;
(b) ;
(c)
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Evaluate the integrals:
(a)
(b)
(c)
Solutions:
(a) ;
(b) ;
(c)
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Find the area of the figure enclosed by the graph of the function , the -axis and vertical lines and .
Solution:
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Find the area of the region between the graph of the function and the -axis for .
Round the result to three significant figures.
Solution:
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Find the area of the region between the curve and the -axis on the interval .
Solution:
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Find the area of the region between the curve and the -axis.
Solution:
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Find the area of the region enclosed by the graph of the function and the -axis.
Solution:
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Find the area of the region enclosed by the curve and the straight line .
Solution:
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Find the area between the curves and .
Solution:
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Find the area of the figure between the graphs of the functions and .
Give the result in the exact form.
Solution:
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Find the area of the region enclosed by the graphs of the functions and .
Solution:
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Find the area of the region enclosed by the graphs of the functions and .
Solution:
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Find the area of the region enclosed by the curve and
the straight line .
Solution:
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Find the area of the region enclosed by the curve and
the straight line .
Solution:
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Find the area of the region enclosed by the curves and
.
Solution:
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Consider the functions and .
(a) Write down the intersection points of and .
(b) Find the area of the region bounded by the graphs of and .
Solutions:
(a) ;
(b)
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Let .
(a) Write down the equation of the tangent to at .
(b) Find the area of the region bounded by the graph of and this tangent.
Solutions:
(a) ;
(b)
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Let .
(a) Write down the equation of the tangent to at .
(b) Find the area of the region bounded by the graph of and this tangent.
Solutions:
(a) ;
(b)
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Let .
(a) Write down the equation of the normal to at point .
This normal intersects the graph of the function at and at another point .
(b) Write down the coordinates of .
(c) Find the area of the region bounded by the graph of and the normal.
Solutions:
(a) ;
(b) ;
(c) Area
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Find the area of the region enclosed by the curve ,
the straight line and the -axis.
Solution:
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Consider the function .
(a) Write down the equation of the tangent to at .
(b) Write down the equation of the tangent to at .
(c) Write down the intersection point of these tangents.
(d) Find the area of the region bounded by the graph of and both tangents.
Solutions:
(a) ;
(b) ;
(c) ;
(d) Area