The area under the curve 𝑦 equals three 𝑥 squared minus 𝑚𝑥 plus six bounded by the lines 𝑥 equals two and 𝑥 equals three is 7.5. Find the value of 𝑚.
Area under a curve can be found using integration. Without knowing the value of 𝑚, we don’t know the exact positioning of this quadratic curve relative to the coordinate axes. But we do know that the area under this curve between the vertical lines 𝑥 equals two and 𝑥 equals three is 7.5. We can express this area then, as a definite integral is equal to the definite integral between two and three of three 𝑥 squared minus 𝑚𝑥 plus six with respect to 𝑥. And as we know the area is equal to 7.5, we can set this integral equal to 7.5 to form an equation. We now need to calculate this integral in terms of 𝑚. And we see that the integrand is a polynomial.
So we recall that to integrate a general polynomial term of the form 𝑎𝑥 to the 𝑛th power with respect to 𝑥 where 𝑛 must be some real constant not equal to negative one. We increase the exponents by one and divide by the new exponents to give 𝑎𝑥 to the power of 𝑛 plus one over 𝑛 plus one plus the constant of integration 𝑐 if we’re performing an indefinite integral. Although, as we have limits of two and three, we’re not concerned about constants of integration here. Applying this rule then and integrating term by term, we have three 𝑥 cubed over three minus 𝑚𝑥 squared over two plus six 𝑥 evaluated between two and three is equal to 7.5.
Now, we can simplify our first term somewhat. Three 𝑥 cubed over three is simply equal to 𝑥 cubed. Next, we need to substitute our limits for this Integral. Doing so gives three cubed minus 𝑚 multiplied by three squared over two plus six multiplied by three minus two cubed minus 𝑚 multiplied by two squared over two plus six multiplied by two is equal to 7.5. Evaluating each of these terms and we have 27 minus nine 𝑚 over two plus 18 minus eight minus two 𝑚 plus 12 is equal to 7.5. Simplifying, we have 27 plus 18 minus eight minus 12 which is equal to 25 and negative nine 𝑚 over two plus two 𝑚 or plus four 𝑚 over two which is equal to negative five 𝑚 over two. So our equation simplifies to 25 minus five 𝑚 over two equals 7.5.
Adding five 𝑚 over two to each side and then subtracting 7.5 or 15 over two from each side gives the equation fire 𝑚 over two equals 35 over two. And as the denominators of these two fractions are the same, we can just equate their numerators, given the equation five 𝑚 equals 35. We then divide by five to find that 𝑚 is equal to seven. So by recalling that the area under a curve between two 𝑥 values can be found using definite integration, we found that the value of 𝑚 is seven.