Explanation Given: The given function of this question is f ( x ) = sin ( m x ) sin ( n x ) The graph: The graph was obtained as follows. Conjecture for I = 0 is For any m ≠ n ; I = ∫ 0 π f ( x ) d x = 0 Formula used : The multiplication of sin A and sin B is given by, sin A sin B = 1 2 [ cos ( A − B ) − cos ( A + B ) ] Interpretation: When we are calculating I = ∫ 0 π f ( x ) d x using a graph, we measure the area under the curve of f ( x ) . In this graph we have two curves of f ( x ) with different values for m and n . First, we consider the curve for m = 2 and n = 4 . There we can see two peaks above the x axis and two valleys lower the x axis. To measure the area under the curve we can get the width and the heights of these peaks and valleys. The widths are nearly two-two and half squares of these four shapes. And the heights are also equal to about three squares. Therefore we can say that the area under the curve f ( x ) = sin ( 2 x ) sin ( 4 x ) is approximately zero because the summation of same areas in opposite values...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 5, Problem 83CRE

To determine

To plot:

The graph of $f (x) = \sin (m x) \sin (n x)$ on $[0, π]$ for the pair $(m, n) = (2, 4), (3, 5)$

Theapproximate value ofeach $I = \int_{0}^{π} f (x) d x$ and a conjecture for when $I = 0$ using the plot.

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Chapter 5, Problem 82CRE

Chapter 5, Problem 84CRE

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Chapter 5 Solutions

CALCULUS (CLOTH)

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To plot: The graph of f ( x ) = sin ( m x ) sin ( n x ) on [ 0 , π ] for the pair ( m , n ) = ( 2 , 4 ) , ( 3 , 5 ) Theapproximate value ofeach I = ∫ 0 π f ( x ) d x and a conjecture for when I = 0 using the plot.

Solution Summary: The author plots the graph of f (x) = sin (m x ) and a conjecture for when I = 0 using the plot.

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To plot:

The graph of $f (x) = \sin (m x) \sin (n x)$ on $[0, π]$ for the pair $(m, n) = (2, 4), (3, 5)$

Theapproximate value ofeach $I = \int_{0}^{π} f (x) d x$ and a conjecture for when $I = 0$ using the plot.

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Chapter 5 Solutions

To plot: The graph of f ( x ) = sin ( m x ) sin ( n x ) on [ 0 , π ] for the pair ( m , n ) = ( 2 , 4 ) , ( 3 , 5 ) Theapproximate value ofeach I = ∫ 0 π f ( x ) d x and a conjecture for when I = 0 using the plot.

Solution Summary: The author plots the graph of f (x) = sin (m x ) and a conjecture for when I = 0 using the plot.

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To plot: The graph of f(x)=sin(mx)sin(nx)on [0,π] for the pair (m,n)=(2,4),(3,5) Theapproximate value ofeach I=∫0πf(x)dx and a conjecture for when I=0using the plot.

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Chapter 5 Solutions

To plot:

The graph of $f (x) = \sin (m x) \sin (n x)$ on $[0, π]$ for the pair $(m, n) = (2, 4), (3, 5)$

Theapproximate value ofeach $I = \int_{0}^{π} f (x) d x$ and a conjecture for when $I = 0$ using the plot.