C2 Area by Integration

Hi all. My logic here is that I do not need to find the integral of the line at all, instead I should be able to just integrate the curve with limits 1 to 3, and I should get the area of the shaded region.

According to the mark scheme this is not the case. Apparently you must (integral of line lim 1 to 3) - (integral of curve lim 1 to 3)

I am failing to understand why, as surely the shaded area here can be found by integrating the curve alone with those limits?

Thanks in advance.

Reply 1

Kevin De Bruyne

21

Original post by PedanticStudent

Hi all. My logic here is that I do not need to find the integral of the line at all, instead I should be able to just integrate the curve with limits 1 to 3, and I should get the area of the shaded region.

According to the mark scheme this is not the case. Apparently you must (integral of line lim 1 to 3) - (integral of curve lim 1 to 3)

I am failing to understand why, as surely the shaded area here can be found by integrating the curve alone with those limits?

Thanks in advance.

The integral of the line alone between x = 1 and x = 3 yields the area bounded by the line from those two points and the x axis. Similar for the curve, which is why you do what you do.

Reply 2

Notnek

21

Original post by PedanticStudent

Hi all. My logic here is that I do not need to find the integral of the line at all, instead I should be able to just integrate the curve with limits 1 to 3, and I should get the area of the shaded region.

According to the mark scheme this is not the case. Apparently you must (integral of line lim 1 to 3) - (integral of curve lim 1 to 3)

I am failing to understand why, as surely the shaded area here can be found by integrating the curve alone with those limits?

Thanks in advance.

The integral between 1 and 3 of the curve gives you the area under the curve between those limits. This means the area that goes from the curve down to the x-axis.

To find the shaded area you need to subtract the area under the curve from the area under the line.

Reply 3

PedanticStudent

OP

13

Original post by notnek

The integral between 1 and 3 of the curve gives you the area under the curve between those limits. This means the area that goes from the curve down to the x-axis.

To find the shaded area you need to subtract the area under the curve from the area under the line.

Ahhhh. I've got you. I assumed that 'area under the curve' meant the area on the inside of the curve.

Thanks! I got into the habit of doing too many of these where the curve is negative, so that is what caused confusion.

Reply 4

PedanticStudent

OP

13

Original post by notnek

The integral between 1 and 3 of the curve gives you the area under the curve between those limits. This means the area that goes from the curve down to the x-axis.

To find the shaded area you need to subtract the area under the curve from the area under the line.