This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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6. (a) Evaluate _0^ ^4 3x 3x \, dx Step 1: Use substitution to simplify the integral. Let u = 3x. Then, differentiate u with respect to x: (du)/(dx) = -3 3x Rearrange to find 3x \, dx: 3x \, dx = -(1)/(3) du Step 2: Change the limits of integration. When x=0: u = (3 · 0) = (0) = 1 When x=: u = (3 · ) = (3) = -1 Step 3: Substitute u and the new limits into the integral. _0^ ^4 3x 3x \, dx = _1^-1 u^4 (-(1)/(3)) du = -(1)/(3) _1^-1 u^4 du Swap the limits and change the sign: = (1)/(3) _-1^1 u^4 du Step 4: Integrate with respect to u. = (1)/(3) [ (u^5)/(5) ]_-1^1 Step 5: Evaluate the definite integral using the new limits. = (1)/(3) ( ((1)^5)/(5) - ((-1)^5)/(5) ) = (1)/(3) ( (1)/(5) - (-(1)/(5)) ) = (1)/(3) ( (1)/(5) + (1)/(5) ) = (1)/(3) ( (2)/(5) ) = (2)/(15) 6. (b) Find the volume of the solid generated when the curve y = x^3 is rotated completely about the x-axis between the ordinates x=0 and x=3. Step 1: Write down the formula for the volume of revolution about the x-axis. The volume V of a solid generated by rotating a curve y=f(x) about the x-axis from x=a to x=b is given by: V = _a^b [f(x)]^2 dx Step 2: Substitute the given function and limits into the formula. Given y = x^3, a=0, and b=3. V = _0^3 (x^3)^2 dx V = _0^3 x^6 dx Step 3: Integrate with respect to x. V = [ x^6+16+1 ]_0^3 V = [ (x^7)/(7) ]_0^3 Step 4: Evaluate the definite integral using the limits. V = ( ((3)^7)/(7) - ((0)^7)/(7) ) V = ( (2187)/(7) - 0 ) V = (2187)/(7) Drop the next question.