**Find the integral ∫∞0e−xdx.**

__Question:__**Options:**

: B

We can see that the given integral is an improper integral as one of its limits is not finite.

In such cases where we have to deal with infinity as the limits of definite integral, we’ll change the limit which is not finite to a variable and then put the limits.

∫∞0e−xdx.=lima→∞∫a0e−xdx

=lima→∞(−e−x)a0

=lima→∞(−e−a−(−e0))

=0−(−1)

=1

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**Answer: Option C**

: C

The given equation of curve can be written as x=f(y)=9y−y3.Now to calculate the area we need to find the boundaries of this curve i.e ordinates or the point where this curve is meeting Y - axis.

f(y)=y(9−y2)

f(y)=y.(3+y)(3−y)

So, the points where f(y) is meeting y - axis are y = -3, y = 0 & y = 3.

Important thing to note here is that the function is changing its signs.

I.e. from y = -3 to y = 0 f(y) is negative.

& from y = 0 to y = 3 f(y) is positive.

Since, the function in negative in the interval (-3, 0 ) we’ll take absolute value of it. Because we are interested in the area enclosed and not the algebraic sum of area.

Let the area enclosed be A.

A=∫0−3|f(y)|dy+∫30f(y)dyA=∫0−3|9y−y3|dy+∫30(9y−y3)dyA=−∫0−3(9y−y3)dy+∫309y−y3dyA=−[9y22−y44]0−3+[9y22−y44]30A=−[0−0−812+814]+[812−814]A=81−812A=812

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