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Question 43968

posted 10 months ago

\text { Evaluate the integral } \int_{0}^{1} \int_{2 x}^{2} 4 e^{y^{2}} d y d x

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Question 43967

posted 10 months ago

\text { Evaluate the integral } \int_{T} 2 x y d y d x, \text { where } T \text { is the triangle with vertices }(1,1),(2,0)
and (2,2).

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Question 43966

posted 10 months ago

a) Sketch the region of integration for the following integrals
\text { (d) } \int_{0}^{\pi / 4} \int_{0}^{1 / \cos \theta} \sin \theta r d r d \theta
\int_{0}^{3} \int_{y+1}^{4} \sin (x y) d x d y
\text { (c) } \int_{2}^{4} \int_{\pi / 2}^{5 \pi / 4}\left(r^{2}+1\right) r d \theta d r
\int_{0}^{5} \int_{-2}^{x+1}\left(x^{2}+y^{2}\right) d y d x

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Question 43970

posted 10 months ago

\text { 1) Evaluate } \int_{-2 \sqrt{2}}^{2 \sqrt{2}} \int_{|x|}^{\sqrt{16-x^{2}}} \frac{x^{2}}{x^{2}+y^{2}} d y d x

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Question 43969

posted 10 months ago

\text { Compute } \int_{R} 15 x^{2} y d A \text { where } R \text { is the shaded region shown below. }

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Question 44066

posted 10 months ago

Points P (ap?, 2ap) and Q (aq²,2aq) where p and q are parameters lieon the parabola with Cartesian equation y2 = 4ax.
OP is perpendicular to OQ where O is the origin.
Show that pq =-4.
The chord PQ crosses the xaxis at point M.
The focus of the parabola lies at point K.
Find the length of KM in terms of a.

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Question 44067

posted 10 months ago

A curve has param etric equations x = 4 sin e and y = 2(1- cos theta) 0<theta<pi/2
Find a Cartesian equation of the curve.
Give your answer in the form y = f(x).
I=\int_{0}^{1} x^{2} d y
The integral / is defined as
\text { Write } I \text { in terms of } \theta \text {. }
Evaluate !.
\text { Find the equation of the tangent to the curve when } \theta=\frac{\pi}{6} \text {. }
Give your answer in the form m/nwhere m and n are integers.

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Question 44712

posted 10 months ago

The expression secxcscx cot x is equivalent to
\sec ^{2} x
\csc ^{2} x
\tan ^{2} x
\sec ^{2} x \tan x

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Question 44715

posted 10 months ago

\text { Detenmine the exact value of } 3 \cos ^{2} \frac{\pi}{8}-3 \sin ^{2} \frac{\pi}{8}

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Question 44716

posted 10 months ago

Algebraically solve the following within the given domain. When necessary, state any restrictions.
\text { a) } \cos \theta(\tan \theta+\sqrt{3})=0 \text { for } 0 \leq \theta<2 \pi
\cos (2 \theta)=2 \sin ^{2} \theta \text { for } 0 \leq \theta \leq 2 \pi

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