Remark
The area of a sector (for degrees) = (x/360) * pi * r^2
What you have to do is take the value for the whole sector and subtract the area of the smaller sector. That will leave you with the shaded area.
Step One
Find the area of the sector with the radius of 4 + 3 = 7
Area = (60/360) *pi * 7^2
Area = (1/6) * pi * 49
Area = 49/6 * pi
Step Two
Find the area of the smaller sector
r = 3 cm
Area = (60/360) * pi * 3^2
Area = 540/360 * pi
Area = 3/2 * pi
Step 3
Subtract step 3 from step 2
Area_1 - Area_2 = 49/6 *pi - 3/2*pi
Area_1 - Area_2 = (49/6 - 3/2)*pi
Area_1 - Area_2 = (49/6 - 9/6)*pi
Area_1 - Area_2 = 40/6 * pi = 6 2/3 pi = 6.667 pi