The figure shows 𝑦 equals 𝑥 cubed minus six 𝑥 squared plus 11𝑥 minus three. Evaluate the area of the shaded region giving your answer as a fraction.
We can see from the graph that actually the limits of the region that we’re looking at are two as the upper limit and one as the lower limit. So therefore, we can see that if we want to find the area beneath the curve between the points two and one, then we can use a definite integral. And a definite integral we’ll want to find is the one with limits two and one of 𝑥 cubed minus six 𝑥 squared plus 11𝑥 minus three. However, this will actually give us the total area beneath the graph between the points two and one.
And what we’re looking to find is actually the shaded region. So what else would we have to do? So we’ll have to find the area of this rectangle here and then actually subtract it from the total area beneath the curve between these two points. So now, we know what we need to do. Let’s do the first stage and actually find the area beneath the curve between the points two and one.
So just to remind us how we actually find the value of a definite integral, we can say that if we have the definite integral of 𝑓 𝑥 between the limits 𝑏 and 𝑎. And this is equal to the integral of our function which I’ve notated using a capital 𝐹 with the upper limit substituted in for 𝑥 minus the integral with the lower limit substituted in for 𝑥. Okay, now, we know what to do, let’s find the value of our definite integral.
So the first stage is to actually integrate our expression. So we’ve got 𝑥 to the power four over four minus six 𝑥 cubed over three plus 11𝑥 squared over two minus three 𝑥. And just to remind you how we actually integrate that, I’m just going to take the first term. So what we did is we had 𝑥 to the power of and then it’s three plus one because you add one to the exponent and then you divide it by the new exponent. So three plus one four, which gave us 𝑥 to the power of four over four. So then, we simplify. So we get 𝑥 to the power of four over four minus two 𝑥 cubed plus 11𝑥 squared over two minus three 𝑥.
So now, what we’re going to do is actually find the value of our definite integral by now substituting in our upper and lower limits into our integrals instead of 𝑥. So what we’re going to get is our upper limit substituted in — so two to the power of four over four — minus two times two to the power of three plus 11 times two to the power of two over two minus three multiplied by two. And then this is minus — and now, we’re substituting our lower limit — one to the power of four over four minus two multiplied by one cubed plus 11 multiplied by one squared over two minus three multiplied by one.
And then if we simplify this, we get four minus 16 plus 22 minus six. And this is minus a quarter minus two plus 11 over two minus three, which is equal to four minus three-quarters, which is equal to 13 over four. So therefore, we can say that the area underneath the curve between the limits two and one is 13 over four.
We now want to work out actually the area of our rectangle, which I’ve highlighted here. Well, we know the area of the rectangle is gonna be the base multiplied by the height. We’ve got a base of one and a height of three. So therefore, the area of the rectangle is gonna be equal to three. So brilliant, we’ve now got the area of the rectangle and the area beneath the curve between the points two and one.
So now, let’s find the area of the shaded region. Well, the area of the shaded region is gonna be equal to 13 over four. So it’s the area beneath the curve minus three, which is gonna be equal to 13 over four minus 12 over four. So I’ve actually changed our three into quarters, which gives us our final answer of a quarter.
And let’s just double check that we’ve actually left our answer in the correct form. Well, we’ve just checked back in the question. It says, leave your answer as a fraction. So great, yes, a quarter is a fraction. So we can say that the area of the shaded region is a quarter.