Video Transcript
Calculate the area of the plane
region bounded by the curve 𝑦 equals 𝑥 squared plus six 𝑥 minus seven and the
𝑥-axis.
Well, first of all, we can see the
we are actually looking for the region that’s bounded by the curve and the
𝑥-axis. Well, we know that actually at the
𝑥-axis 𝑦 is going to be equal to zero. So therefore, the first thing we
want to do is actually find the roots of a function. So therefore, in order to do this,
what we can actually do is set our function equal to zero. So we have 𝑥 squared plus six 𝑥
minus seven equals zero.
So then, next to find our roots,
what we do is we actually factor our quadratic. So we get 𝑥 plus seven multiplied
by 𝑥 minus one. So just to remind us how we
actually found that, we got two numbers that are actually multiplied together to
give us negative seven. So positive seven and negative one
give us negative seven. And then two numbers need to add
together to give us positive six. So positive seven plus negative one
gives us positive six.
Okay, so now let’s actually find
our roots. So therefore, we can say that 𝑥 is
equal to negative seven or one. And we actually found this because
to find out what our roots would be we just set each of our parentheses equal to
zero. So if our first parenthesis was
negative seven plus seven, this gives us zero. So zero multiplied by anything
gives us zero and for our second parenthesis, one minus one gives us zero, so again
giving us a zero for the function.
Okay, so now we’ve got our
roots. What do we do next? So now, so what this is [inaudible]
to do is actually now draw a sketch of our graph. So we’ve actually got a sketch of
the curve 𝑦 equals 𝑥 squared plus six 𝑥 minus seven 𝑦. And you can see the roots on the
𝑥-axis. And we know that the shape is a
U-shaped parabola because actually if you look back, we’ve got a positive 𝑥 squared
term. So therefore, we know it will be
U. If it was a negative 𝑥 squared, it
would be an inverted U.
Okay, so now, let’s look for which
region now we’re trying to find. Well, we’re looking for this region
here because we’re looking for the region that is actually bounded by the curve 𝑦
equals 𝑥 squared plus six 𝑥 minus seven and the 𝑥-axis itself. But how do we actually find the
area of this region? Well, we actually find the area of
this region using a definite integral. So we can say that the definite
integral of our function with the limits 𝑏 and 𝑎.
But let’s consider why that is. So actually, if we have a small
section here that I’ve marked in blue, so we can say the width of our rectangle — so
the small section forms a rectangle — would be 𝑑𝑥. So it’s very small proportion of
𝑥. And our height to that point will
just be equal to our function — so 𝑓 𝑥. So therefore, if we want to find
the area of our rectangle that we’ve drawn, then what we do is actually multiply our
two sides together — so 𝑓 𝑥 our function multiplied by 𝑑𝑥. Well, this would just give us the
area of one little rectangle.
However, if we want to find the
total area of our region, what we’d need is lots of these little rectangles. And what definite integral does it
is that actually gives us the area of an infinite number of these rectangles. We say an infinite number because
if it was any other finite number, then therefore we’d just be estimating. But actually, it gives us the
complete area of this region.
Okay, so now we know what we need
to do. Let’s use it to find the area of
the region. So we now know that the area of the
region is gonna be equal to the definite integral of 𝑥 squared plus six 𝑥 minus
seven between the limits one and negative seven. Okay, but how do we actually
calculate a definite integral.
So to find a definite integral,
what we can say is that if we’re looking for the definite integral of a function
between limits 𝑏 and 𝑎 then this is equal to the integral of our function with 𝑏
substituted in for 𝑥 minus the integral of our function with 𝑎 — so our lower
limit substituted in for 𝑥. Okay, so let’s use this to find the
value of our definite integral.
So the first stage is to actually
integrate our function. So we get 𝑥 cubed over three plus
six 𝑥 squared over two minus seven 𝑥. And just to remind us how we
actually got that, we’re going to look at the second term. So if we were going to integrate
six 𝑥, then what we get is six 𝑥 and then to the power of one plus one cause you
actually add one to the exponent. So that gives us six 𝑥 squared. And then, you divide by the new
exponent — so you divide by one plus one, so you divide by two, which gives us six
𝑥 squared over two.
So then, we actually simplify and
we have 𝑥 cubed over three plus three 𝑥 squared minus seven 𝑥 and then that’s
between the limits one and negative seven. Okay, so now let’s move on to the
next stage.
So now, the next stage is to
actually substitute in our limits for 𝑥. So we’re going to get one cubed
over three plus one multiplied by one squared minus seven multiplied by one and then
this is minus negative seven all cubed over three plus three multiplied by negative
seven squared minus seven multiplied by negative seven which is going to be equal to
negative three and two-thirds minus negative 343 over three. So be careful here, a common
mistake if we got negative seven cubed, it still remains as a negative value which
is negative 343 plus 147 plus 49 that’s because we had negative seven multiplied by
negative seven which is a positive, which is all equal to negative 256 over
three.
But it’s negative, is this
right? Well, actually, we can see that the
region we’re looking at is actually below the 𝑥-axis. So yes, we would expect it to give
us a negative value because as I said it’s below the 𝑥-axis. So therefore, we can say that the
area of our region bounded by the curve 𝑦 equals 𝑥 squared plus six 𝑥 minus seven
and the 𝑥-axis is equal to 256 over three area units. And we’ve taken away the negative
cause actually we’re just looking at the area of that shape. So it’s the area is 256 over
three.
