Question 1
Find the area of the finite region in the first quadrant bounded by the graphs of the equations
![](data:image/png;base64, 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)
and
![](data:image/png;base64, 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)
.
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Question 2
Find the area of the finite plane region bounded by the graphs of the equations y = cot x, y = 0,
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAAmACQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD9U6K4fw78afCHivUF0/S9QubjUvts2ny2DabdR3NtNEoZ/PieMPAm1oyJJAqN5sWGPmJuTxj8dvhr8O9Y/snxX8Q/CnhjVfLWb7DrOt21pPsbO1tkjq204ODjBwaV1ZPvsOz2OzvLuHT7Oe6uHEdvBG0sjkZ2qBkn8hXlWnftS+A9U1GwsoY/F8c19dR2UDXPgbXIImmdQyIZJLNUXKkNliAF+Y4HNemaFr2meKNHtNW0bUbTV9KvIxNbX1jOs0E6HoyOpKsD6g4rzjxF8HdcvYvDd3ovi9NM13R/EN1rzXN7pzXdrdidLiJ4HgE6MAsVxtRlkG3YDg5Iq425ve2/r+v0Yrpq6/rT+v8ANHq1FFFSB5H4f+CWr6B8SYvHaeLIpNf1BXtvEaf2YRa6nbAk2yRR+cTbPCMKr7n3AvvViVKdH4x+DegeOtY/tPUtQ8V21z5axbNG8X6tpkGBnB8m1uo493PLbcnjJOBXc0UrLRdht3bfc5DWfhH4O8UeEtO8M+JPDtl4v0SwZZLe28Up/a5EiqyrIz3Rkd5AruN7Ethjzya8g1D4Q/Aiz+L2ieBI/gX4Enl1DT7y9k1D/hGrARwNbm3/AHWPJyzEXKMeRtBXru4+jq+ax8BfiTpX7SXhvxbZeK31XwTp51AzwanqNnFdf6Y0TyokMOkDKK0Sj57kuwVMPGFZX0puPOlPaz++zt+NgSSi32PonSdJsdA0qy0zTLK307TbKFLa1s7SJYoYIkUKkaIoAVVUABQAAAAKKt0VAgooooAKKKKACiiigD//2Q==)
, and
![](data:image/png;base64, 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)
.
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![](data:image/png;base64, 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)
ln 2 square units
◦ ln 2 square units
◦ 2 ln 2 square units
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![](data:image/png;base64, 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)
ln 2 square units
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![](data:image/png;base64, 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)
ln 2 square units