1- find the volume of the solid obtained by rotating the region under the curve
ID: 2845441 • Letter: 1
Question
1- find the volume of the solid obtained by rotating the region under the curve y= 1/root (x+1) , from 0 to 1 about the x- axis ?
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2- Prove the third law of logarithms. [ hint : start by whoing that both sides of the equation have the same derivative.] ?
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3- show that the function y= e^(x)^2) erf(x) , satisfies the differential equation : y' = 2xy + 2/root (pi)
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4- (a) show that e^x > or equal 1+x if x > or equal 0.... [ Hint : show that f(x) = e^x- ( 1+x) is increasing for x > 0 ]
(b) Deduce that 4/3 < or equal the integral ( 0 to 1) e^(x)^2 dx < or equal e
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5- example :::: (a) by comparing areas , show that 1/2 < ln2 < 3/4
(b) Use the midpoint Rule with n= 10 to estimate the value of ln 2.
refer to the example :
(a) find an equation of the tangent line to the curve y=1/ t , that is parallel to the secant line AD
(b) Use part (a) to show that ln2 > 0.66
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6- Let
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please explain with the answer , ( i will add mor epints if the answers were clear )
please be careful when u answering # 5 the answer should be for the secound (a) and (b) ..
Explanation / Answer
1.
to find the volume, we need to integrate the area as the follwing:
A(x) = ? x^2
A(x) = ? [ 1/sqrt(x+1) ] ^2
A(x) = ? [ 1/(x+1) ]
1
? ? [ 1/(x+1) ] dx
0
...................1
? * ln ( x+1 ) ] = ? * [ ln(1+1) - ln(0+1) ] = ? * [ ln(2) - 0 ] = 2.18 unit^3
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