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Differentiate the given trigonometric functions.
1. π(π₯) = 3 π ππ2π₯ πππ 4π₯
2. π¦ = 1/2 π‘ππ^4(π ππ 6π₯)
3. π(π‘) = tan π‘/πππ π‘β4
4. π¦ = 8 ππ π π₯/4 πππ‘ π₯/4
5. π(π₯) = π₯π ππ3π₯ + π₯ πππ 3π₯
1.Β Β Β Find the volume generated under the curve y=x^3 + 3 and the x-axis between x=0 and x=3, rotated about the x-axis.
Integrate the following functions w.r.t. x using substitution:Β
Β i) x x 1dx
1
3 4
β«
+ ii) β«
Ο
4
7 2
tan x sec x dx
Β iii) β«
sec (cos )x sin x dx
The electromotive force for an electric circuit with a simplified generator is πΈ(π‘) π£πππ‘π at π‘ π ππππππ , where πΈ(π‘) = 50 π ππ120ππ‘. Find the instantaneous rate of change of πΈ(π‘) with respect to π‘ at (a) 0.02 π ππ and (b)0.2 π πc
2. \:\int _0^4\int _0^{\sqrt{y}}\:xy^2\:dx\:dy
3. \int \int R\:\left(6x^2y^3-5y^4\right)\:dA,\:R=\left\{\left(x,y\right)\left|0\le x\le 3,0\le y\le 1\right|\:\right\}
4. \int \int R\:xye^{x^2y}\:dA,\:R=\left\{\left(x,y\right)\left|0\le x\le 1,0\le y\le 2\right|\:\right\}
5. Find the volume of the solid that lies under the surface z = xy and above the triangle with vertices (1, 1), (4,1), and (1, 2).
6. Evaluate \:\int _0^1\int _{x^2}^{\sqrt{x}}\:x^2ydydx and find the volume.
surface area A increases as b increases.
