Region D in xy-plane is bounded by 4 lines-2x+y=0-2x+y=pix+y=0x+y=pi/2integralD sin(y-2x)cos(x+y) =?

Question

((CALCULUS 3))

Consider the region D in the xy-plane bounded by 4 lines:
-2x+y=0
-2x+y=pi
x+y=0
x+y=pi/2
Determine the integralD of sin(y-2x)cos(x+y)dxdy

Answer 1

Given the four lines

-2x + y = 0
-2x + y = π
x + y = 0
x + y = π/2

Determine the area bounded by the lines.

First observe that the first two lines have the same slope.
m1 = 2

And the last two lines have the same slope.
m2 = -1

So we are finding the area of a parallelogram.
______

Find the points of intersection of the lines.

-2x + y = 0
x + y = 0

By inspection these two lines intersect at the origin O(0,0).
_____

-2x + y = 0
x + y = π/2

Subtract the first equation from the second.

3x = π/2
x = π/6
y = 2x = π/3
P(π/6, π/3)
________

-2x + y = π
x + y = 0

Subtract the first equation from the second.

3x = -π
x = -π/3
y = -x = π/3
Q(-π, π/3)
__________

OP = <π/6, π/3, 0>
OQ = <-π, π/3, 0>

A = Area of parallelogram

A = || OP X OQ || = || <π/6, π/3, 0> X <-π, π/3, 0> ||

A = || <0, 0, 7π²/18 || = 7π²/18

Answer 2

draw graph of the lines, see the region.
Find the intersection points.
Area=1/2*(pi/2)^2-2 integral x=0 to pi/6 integral y=0 to pi/3 of sin(y-2x)cos(x+y)dxdy

Answer 3

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